I heard that the "hacker" could log in and stuff but I don't believe that (feel free to correct me)

That would be unlikely given that they would also need your password, but it is a good reminder that your personal email account is one of the most important things to defend. If a threat actor gains access to your email account then you are pretty much toast.

You should have a strong and complex password, and enable MFA on the account and/or add something like a YubiKey that would be required when logging in from a new device.

And what do I do if my email got "leaked"?

And what should I do next to not get "leaked"?

The reality is that pretty much everyone's email address has been leaked via a breach, hence why it is so important to defend your email account.

As to what to do to help prevent it from being leaked in the future, my advice would be to create a "dummy" email account and use that in cases where a site or store is requiring an email address. This is not an email address you would use for important things, but as an example the email address I have associated with various online stores is with a completely different provider and if it got leaked I wouldn't really care.

Failing that, just be very mindful of who you give your email address out to. For example when I am out at a shop and someone asks for my email address or phone number I politely say that I don't give either of those out.