The PHB p195 (Escaping a Grapple) describes how to break free from the condition but, once free, I assume the grappled creature stays where they are, on a threatened square and therefore attack of opportunity rules apply.
However if the grappler was humanoid and it grappled you with their arms/limbs, I can't imagine how they could attack you when you are escaping because in order to grapple you they would have to sheathe or drop their weapons and, therefore, can't attack you with anything other than their limbs (bludgeoning 1 damage) when you try to leave.
Best Answer
A grapple only takes one hand
So, after they grapple you, and trip you so you can’t stand up because the grapple reduces your speed to 0, they can then wail on you with that long sword they are holding in their other hand, with advantage because you are prone. Why do you think they started the grapple in the first place?
If you break the grapple, your speed is restored and you can then move. They still have that long sword of course, so if you move without Disengaging they’re going to use it to cut you with an opportunity attack. If you are smart, you stood up first so they don’t have advantage but you can only move with the half of your movement you have left, so about 15 feet. You know they’re coming to grapple you again on their next turn, right?
Of course, breaking the grapple is problematic: they almost certainly started it because their Strength (Athletics) is better than yours, or your Dexterity (Acrobatics). Smart enemies don’t grapple fighters or rogues: they grapple wizards or warlocks or other wusses.
I was DM for a party in Tyranny of Dragons who worked out that the best way to kill a dragon was for the Bard to polymorph the Barbarian into a Great Ape who has a much better grapple ability than any dragon, grapple the dragon whereupon the Sorcerer would drop a Wall of Fire on it (cold side to the ape) plus other non-concentration stuff, the Cleric would use Cloud of Daggers plus other non-concentration stuff, the Bard would stay out of the way and the Barbarian would rage and hit the dragon - 2 rounds maximum.
Notwithstanding
An unarmed strike has a reach of 5 feet anyway ... pray it is not a monk grappling you.