The scenario will need some adaptation. Your player is now trained to think about these hard-to-hurt monsters as too powerful for him and his characters to deal with.
It's actually a good sign that he runs away when he feels outmatched (see How can I make my PCs flee? for the flip side of your problem).
One system-specific possibility is that he has missed some option which is assumed by the designers. An extreme example is a party of melee only characters against ranged monsters with flight. Take a look at this and see if there are easily available options to deal with the common monster defenses. Make this an extra sidequest so it's not just a GM handout.
In general, there are two feelings a player needs to get involved in a conflict. The first is confidence that he can find a way to defeat the enemy without losing horribly. The second is investment in winning.
There is a balance between these two elements. The more investment a character has (the monster is threatening an NPC he really likes for instance) in winning, the lower the confidence he needs in winning to try. This also applies in reverse. The more confident he is in winning easily, the less investment he needs to get involved.
Confidence is increased when you achieve something a little harder than you expect to be able to do. The classic psychological text on this is Flow by Mihaly Csikszentmihalyi. If you want to train someone up to feel capable of handling monsters that put out these signals, start with much much weaker ones and show farmers/townspeople/NPC nobody's dealing with them. Then build up the opposition until they know how to handle the ones you care about or have worked out the tools they need.
In the specific example of the Eerie, you need to signal that all is not quite right with it. Give someone a skill check, Knowledge, Nature Lore, Perception, whatever, to see that it's not exactly what it seems. That should give a little boost.
Building investment is a big task. Creating emotions and How can I increase tension during roleplaying? have more advice to give on this subject.
Hint it and measure enthousiasm
The general approach I take when I'm not sure what my players like, or whether they'd enjoy a specific thing, is to hint to it during the session and see if they bite. This works best in an open world or if you've already taught your players that they can say "no" and the story will go on, but even if they're used to being railroaded around you can still drop hints to events and see if they bite.
The basic idea is to include the direction you'd want to take in something that is obviously not directly related to what the players are doing. Commonly used for these purposes are rumours, idle chatters, news reels playing in the background, headlines in the paper, etc.
Think of how in movies there is often a movie-set on while the main character is leaving to go somewhere and then just after he slams the door, the camera lingers for a bit and lets you hear the last part of a crazy news report about "possible aliens spotted" or "scientist does crazy invention; is world doomed?" or what have you.
You can have those in your game, too. The next time the players are in a bar, describe how the radio is on and they overhear how a missing kitten was found, a 100-year old man was visited by the mayor, and a resident crazy person claims their best friend was eaten by a dusty book they found in the attic. See whether the players remark on the last story and whether they think it's interesting. (This can be in character or out; either means it has peaked their interest)
If you get an initial reaction, bring that story, or elements of it, back later. Keep them in the background, but let the players get a glimpse of how it's going. Make it sound like whoever is dealing with it, could use a hand. But don't make it sound like they ought to get involved yet. Again, measure their reaction to it. If they have at least something of a reaction, you can drop the story more directly into the game. Have them meet someone who is part of the investigation, who tries to contact them for help. Preferably while they are in the middle of something. Let the tone simmer through in the way this request is formed; this'll be the first time that you really present the tone you're aiming for, but while the players are busy.
If they immediately drop everything to help this person; they are really interested in changing tone. Go wild. If the players sound enthousiastic, but the characters want to finish their current job first, they are also interested in the new tone, but make the change more slowly, so they can get used to it. If the players sound confused or seem to accept more out of an "oh, this will be the next story" line of thought, they are probably not very interested and I would consider running the core of the story in your original tone, or just dropping it altogether. These players might enjoy the tone, but not in the current game.
If they flat-out refuse to help or protest out of character, they probably don't want to run in the given tone at all and you should probably drop it and maybe apologise for bringing it up.
However
I feel this is also important to mention. While the above makes it possible to change the tone of a campaign slowly, I would not actually advise doing it. (It's more suited to introducing new topics and stories).
The reason here is that (most) players design their characters and background and set their expectations based on the originally described tone of the game. Characters grow in the direction of the original tone, too. But your new tone would probably be better suited with completely different kinds of characters.
For example, if I know there's going to be a lighthearted cyberpunk game, I might run a glitching cyborg, a tough jock who's afraid of talking to robots, or a street-rat who mugs rich people with a robotic monkey-sidekick. Light-hearted, fun characters to play.
But if you suddenly start changing the tone to something creepy and horrific, these characters would probably not be appropriate.
If you wanted to run a cyberpunk chtulhu game, I'd much rather have heard beforehand, so that I could have created a android infected with a virus that interferes with his motor controls and occasionally his judgement, an ex-army guy who has severe paranoia after nearly being killed by a friendly drone-trooper over a joke, or an orphan who has been without a community so long, he thinks his mindless automaton is actually his friend.
Characters suitable for such kind of game, for whom I'd have explored the background that lead them to where they are and whose mannerisms are designed around a serious theme and for whom I've thought about the horrible fate probably awaiting them.
(Note that the first and second examples are intentionally the same characters, just described for a different kind of tone of campaign. The tone matters hugely in how people portray characters, which parts of their background they write out and what goals they set for them. I wouldn't enjoy losing that work and having nothing to fall back on because the tone switched without my knowing.)
Best Answer
Make the townsfolk afraid of them, too.
In short: have the NPCs react like real people.
It's a small town. People talk. People are going to notice the destroyed door, and others may have been drawn to the ruckus and seen/overheard what happened. Then the rumor mill got to spinning.
See, in this case...
So...in rolls this band of strangers, wipes out their old oppressors, then kicks down the door to their townmaster's home, terrifies him and his daughter, and (depending on what was said) may have just informed him that they are seizing control of the town.
Great.
So, by the perspective of the average townsperson, they've just traded one tyrannical group of thugs for another that's even stronger.
So, this could play out in a few ways...
Assuming the party notices what is going on, Sildar or anyone else who somewhat trusts them can give them an extremely short line, something like...
No need for an extensive lecture
Fixing it
The PCs have made a really, really bad first impression on the town. It's going to take a bit of effort to convince them that they are, in fact, the good guys and not a rival group of thugs that just took over some territory from the Redbrands. Public apologies, repaying protection money, distributing loot from the hideout, etc. may help with this. As will time, with the PCs doing quests and such to help the town.
How it went when I did something like this
Not with LMoP, but I've done stuff like this in the past. When a PC party is too heavy-handed and forgets that NPCs are people, I tend to draw in reactions like this. The town ends up afraid of them, people start avoiding them, and generally treating them like a gang of unstable violent thugs. "Don't provoke them, but try to get them to go away" is the name of the game.
This usually gets the point across to my players without having to lecture them. It's a case of 'show, don't tell.' You show them the consequences of their actions instead of just telling them about it. I've had one group of players who ended up just rolling with it, and ruling a town as a band of little tyrants (and that complicated life for them in several ways throughout the adventure), but they liked their rulership enough that they didn't care. But, by and large...NPC reactions to PCs tend to be a good indicator to players of how their characters' actions are being perceived.