It means that you get an automatic +1d8 radiant damage from Improved Divine Smite, always, on all your melee attacks. You can also still voluntarily use Divine Smite itself to add +Xd8 radiant damage (depending as usual on how many spell slots you spend on it) to a chosen melee attack.
In other words, Improved Divine Smite doesn't replace Divine Smite and doesn't prevent you from using them together. That's all it's saying in the part you have emphasised.
Yes, if the racial rule calls them weapons
(This answer has been updated to reflect the apparent shift in the wording of the rules since Explorer's Guide to Wildemount. The original answer is below the horizontal rule.)
User Mavoc points out that the wording for most of the races with natural weapons has been revised. The online version of those races (and printed versions since Explorer's Guide to Wildemount) uses slightly different wording from previous publishings:
Your talons are natural weapons, which you can use to make unarmed strikes.
Further, it seems this wording change "which you can use to make unarmed strikes" has ben retroactively applied to virtually all races that have natural weapons on D&D Beyond.
For context, the PHB (page 119) states:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).
So, while general rule that unarmed strikes can't work with Improved Divine Smite because unarmed strikes aren't weapons, the rule for races like the Aarakocra, specifically says that the relevant anatomy is a natural weapon. This would allow it to work with abilities that require a weapon despite the fact that an unarmed strike is being made.
As your other linked answer states, Unarmed Strikes do not work for Improved Divine Strike because the rules say Unarmed Strikes are not weapons:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).
But the racial ability for Minotaur horns specifically says the horns are a weapon:
Your horns are natural melee weapons
What is happening here is that there is a specific rule/allowance for the Minotaur that is similar to (but different from) a general prohibition.
The minotaur is not making an Unarmed Strike (a head-butt). It is making an attack with a weapon which happens to be located on its head.
Contrast this with the Aarakocra's Talons:
Talons
You are proficient with your unarmed strikes, which deal 1d4 slashing damage on a hit.
(Hat tip to Justin for the Aarakocra example)
Also, Jeremy Crawford has tweeted an affirmation of this answer.
Best Answer
RAW: yes
Intended or not, the rules define natural weapons as weapons.
So it is simple, natural weapons are, as their name implies, weapons that happen to be naturally occurring on a monster or creature's body. They are used to make melee weapon attacks, they are called weapons in the rules, and they have "weapon" as part of their name. Intended or not, the rules define natural weapons as weapons.
Magic weapon works on any weapon
The spell doesn't put any limits or restrictions on the weapon that can be affected and thus can affect natural weapons as well.
The Sage Advice Compendium says yes.
The Sage Advice Compendium states:
Rules as Intended(?): No
Jeremy Crawford has said in an unofficial tweet:
It's hard to say why he believes that because he does not explain his logic. He seems to imply that the word weapon only applies to the weapon tables in the PHB. However this is a very strange take given that there are many weapons listed in the DMG that are not in this table. They are all magic weapons and thus not eligible to benefit from this particular spell, but they are weapons regardless. So his claim that any mention of "weapon" refers exclusively to the PHB table seems extremely flawed.
However, it is possible he means that this was the way the spell was intended to work.
Honestly, as a DM, I have no idea how I would go about justifying his ruling if I wanted to implement it at my table.
Without more explanation it is unclear, but it seems very clear that what he says is at odds with the RAW.
As always of course, your DM can choose to rule as they please regardless of RAW/RAI.