Yea, I was confused too. It's important in two ways. One is that EX pokémon normally can't be evolved, so this ability lets Evee EX evolve. You don't normally notice the rule since EX pokémon are typically fully evolved pokémon that can't evolve anyway.
The second part is that you can't typically run more than 2 of the same pokémon in a deck, and pokémon typically only evolve from one type of pokémon. So typically you could not evolve Evee EX into Jolteon, since Evee and Evee EX are two different species of pokémon and Jolteon only evolves from Evee. This ability however lets you do exactly that.
Now you can run 2 Evee and 2 Evee EX in the same deck and have all 4 of them evolve into the same Jolteon, Flareon, etc.
Edit: Also, the line "if you play it from your hand onto this Pokémon" means that it won't work with some trainer cards, like those that allow you to pull an evolution card from the deck and play it directly onto the Evee EX.
I'm pretty sure it is just how evolving cards typically works, except that you can't usually evolve ex cards. So it's an ex card that you can specifically evolve.
The only other ex we have that isn't fully evolved is Pikachu ex - and you cannot evolve it into a Raichu during battles.
Normally you can't evolve a card the same turn you put it in play. This card is able to be evolved as soon as it is set, except on the first turn since that would be too OP.
I was wrong, apparently it is just normal evolution.
There are evidently a number of pokemon that have an actual deevolution attack, as well as a deevolution spray you can use on your own pokemon.
So, if this Eevee gets hit with a deevolve attack, then you can play another Eeveelution on your next turn.
Or, you could strategically deevolve your Eevee, and then reevolve it with maybe a different Eevelution that is a better type counter to whatever you are up against now.
You'd be able to do that, and attack, all in one turn, if you already had the Eevee/lution down, and a deevolve spray and alternate Eeveelution in your hand.
Normally, deevolving is devastating, because.... you don't remove damage tokens, but you now just have less HP, so, very often this results in a KO for a moderately beat up, evolved pokemon.
But this base Eevee has 90 frickin HP, which is quite a lot...
I have not played the TCG since I was in middle school, but the OG TCG Eevee only had uh... 40 or 50 HP, from some of the first expansion packs, 40 HP for Team Rocket, 50 HP for Jungle.
...
So yeah, thats my read of this, it is intended to be used probably mostly with a deck that is significantly focused on having this kind of Eevee and its evolutions as a kind of... multi tool, most basically for gaining optimal type attack bonuses, possibly also even more complex if there are some Eeveelution cards with more exotic modifiers or attacks.
It is basically also a decent counter to anyone using deevolve attacks... as this unique trait is paired with an Eevee with a stupid high amount of health for a base form Eevee.
EDIT: have edited this multiple times, took me a bit to fully make sense of it.
Let me go throw it in a deck and report back. I could've sworn when I was doing the NPC battles they were evolving it as soon as it was set, so I assumed it was just a weird translation or confusing wording.