There are really only two ways that your credit limit will be increased on a standard credit card:
You ask for it
Most of the time this means you'll see a new hard inquiry on your credit report. Expect to see a ding of a few points from your credit score (single digits). If your credit is already good, you've been managing the account well, and haven't already had any increases in the last year or two, it'll probably be granted. The hard inquiry stays on your report for 2 years whether or not they grant you an increase.
It's just given to you
If you have been using the card responsibly, the company may give you an increase without you having to ask. You won't see a hard inquiry added to your credit report in this case, but they should write to you to let you know about your new limit (they'll want you to start using it after all - more merchant fee income for them). If you don't want the new limit for some reason*, you'll have to ask them to reduce it. There is no penalty for doing so.
* E.g. you're mortgage shopping and want to minimize your current debt-to-income ratio.
When you say you're "eligible for an increase" I assume you've received a mailer from your credit card company suggesting you ask for one. Even if you are pre-screened, following up on this and asking for the increase will still put a hard inquiry on your report, however the pre-screen means they've already done a soft inquiry and are pretty confident you will qualify.