In the previous Fallout games, not anyone could use a Power Armor. You needed a specific training to be able to use it.
Is there a lore explanation on this?
The primary male character, Nate, is a US Army veteran. It's reasonable to think that he had power armor training while in the service. He's even shown in a unit with other power-armored soldiers in the opening cut scene.
There's no in-game explanation of how anybody else in the game would have power armor training, other than the members of the Brotherhood of Steel.
However, we can speculate.
Given how comparatively common power armor is in the Commonwealth versus other Fallout locales we've seen, it's reasonable to think that the general populace would have had more opportunity to learn how to use them than in areas where all the armor was confiscated by the Brotherhood. It's explained in the loading screen tips that Raider bosses have salvaged and repaired their suits, so of course they would have had to figure out how to use them. The knowledge could have then spread throughout the Commonwealth society over the centuries since the Great War.
All that being said, it doesn't really explain how Nora would be able to just jump in the first suit she sees and defeat a deathclaw. Maybe the shooting ranges of her time had suits the public could try out, or perhaps Nate figured out a way to let her try one while he was in the service.
A Note on Previous Games
The main differences between Fallout 4 and the previous games in this regard are that, unlike the Lone Wanderer or the Courier, the Lone Survivor doesn't need training, and that other NPCs will sometimes use armor without you explicitly giving it to them and telling them to use it.
As noted on Nukapedia's Power Armor Training page, in Fallout New Vegas the player character was the only person that actually required Power Armor training. Followers and NPCs could equip it if you gave it to them. In my head-canon, this was because the player character showed them how to do it, but this doesn't work if you equip a follower with power armor prior to learning how to wear it yourself.
The Nukapedia page on Fallout 3 Companions indicates that the situation was the same in Fallout 3:
All human and ghoul non-player characters (so all except Fawkes, RL-3, and Dogmeat) are capable of wearing power armor.