This is actually a two-parter.
1) The expression 'jugar a' means to play at a sport.
2) When you use this expression, you also have to put an 'el' or 'la' in front of the sport. And whenever you have ['a' + 'el'], they almost always combine to form 'al' instead.
So in effect, the difference between "No me gusta jugar béisbol" and "No me gusta jugar al béisbol" is a slight emphasis with the latter: "I don't like to play baseball" versus "I don't like to play at baseball."
(Translated literally, "I don't like to play to the baseball." Which is an example of why word-for-word translations aren't always a good idea).
Spanish has two required contractions: al ['a' + 'el'] and del ['de' + 'el'].
This helps to distinguish between ['a' + 'el'] and a él, and ['de' + 'el'] and de él.
The only time you don't contract, is if the 'El' is part of a name:
¿Conoces a El Zorro?
Soy de El Salvador.
I'm confused by why I need 'A' on a different question but not this one. When I answer the other it requires you to write, "A mí no gusta jugar al béisbol." However, this one doesn't? Are both correct then? Before anyone says it, I understand the whole adding 'A' for personalisation.
MiHacienda - Some action words &/or phrases require ME, TE or SE to express the action. Also LO is used for some.
ME = I (do it) TE = You (do it) SE = He/She/They (do it) "SE" is used for many things in Spanish.
LO = It (does it) - although this one is used a bit different.
Here is a good link to help explain the details. :)