How you use this material is ultimately up to you and will depend on how you structure your site. If you are basically just providing a stopping point for fans to come and obtain this information, then use it as is and attribute the information to the team. Below is an example on how you might do this.
Press Release - 6/16/2012 - Baltimore Orioles
If you choose to write your own stories based on this information, then you should still provide attribution so that readers know the source of your information. These press releases and other information are just sources, and as such should be acknowledged. One way to do this would be to provide a closing tagline such as this:
This article written with the assistance of information provided by the Baltimore Orioles.
If you were actually at the game or heard ot in the radio or something, then you might mention that in your article, but that would the only case in which attribution would not be directly necessary.
You should also have a copyright notice on your site that states that all information pertaining to each team is the property of MLB and may not be reproduced without their express written permission. (MLB.com will have more specific guidelines.)
This of course brings up another point. Have you yourself obtained that "express written permission"? I used to be a co-owner of a fantasy baseball site, and over ten years ago MLB became the sole owner and executor of ALL things related to professional baseball in the United States and abroad. At the time our site was active, we received promotional materials straight from the actual teams, and we then gave those materials as prizes. We received a notice from MLB that we could no longer use the logos, team names, team statistics, or any promotional materials for any specific team without first agreeing to a licensing contract. We also received a notice from the MLBPA (player's association) that we could no longer use a player's name or likeness, his individual stats (which were now fully owned by MLB), or even his uniform number in reference to a specific team without signing a licensing contract with them AND MLB!