Why not " I would like to play." Would that be something different?
That would be closer to «Я хотел(а) бы поиграть».
I want to play sounds like a spoiled child throwing a tantrum.
Russian sentence sounds like this too.
If you roll the «Р» right, then «играть» kinda sounds like the Danish/Nordic word "idræt", which is a tough word to translate, but would be something between "physical education" and "gymnastics, ball game & sport". It's also a school subject, and there is some playing, going on. I know there's a difference on "R" and "D", however there can be similarities when rolling the "R". "R" is even pronounced like "D" in Japanese, sometimes.
So what I find interesting is that «играть» means "to play", and that the cross-country homophone "idræt", has in some way to do with 'playing'. But several sources say that it's a domestic word to the Nordics (not known anywhere else), and that the origin of the word is contentious.
It makes me speculate that the words «играть» and "idræt" may be false cognates, false friends or have a have common ancestor.
Or possibly just a silly coincidence :)
I don’t know Danish, but when I listen to the word pronounced on the web, the main difference is not the but the in place of <г> in играть. There are minor ones in the pronunciation of the sounds and the final which probably isn’t always “soft” (palatised) in Danish as it is in Russian. But the [d] vs [g] one is important because both languages distinguish those sounds. Also I don’t know of any confirmed cases where Slavic has [g] in place of Germanic [d], so my instinct tells me the similarity is probably a coincidence.
Wiktionary also points in that direction: It cites “idræt” (and its cognates in other Scandinavian languages) as coming via Icelandic from Old Norse ”íþrótt” > ið (effort) + þróttr (strength). Играть on the other hand seems to come from a Proto-Slavic root *jьgra- (which seems to have had a meaning at least close to “play” already) + an infinitive ending. So both come from two parts, but in the Scandinavian word basically only the i- is one part and the rest is the other, whereas in the Russian word all of игра- is one part and -ть the other. That makes it very unlikely that they are related, unless maybe extremely remotely if i- and игра- come from the same Proto-Indo-European root, but I doubt it. Almost certainly only a coincidence ;)