To "reserve one's rights" is to make it clear that some other action is not intended to give the rights up. Typically, it has no legal effect but is used out of caution, to avoid misunderstanding.
For example, an author who publishes their writing for free online might add "all rights reserved" to the copyright notice. This makes it clear that by making the work freely accessible, they did not intend to give others permission to republish it, or release it into the public domain.
Similarly, at an early stage of a legal dispute, a lawyer might write a letter of demand that outlines some of the client's strongest points, without wishing to imply that the case has been fully investigated and pleaded. Such a letter might "reserve our client's rights" to add to or amend the claim later.
In both cases, the legal situation would probably be the same if the rights were not expressly "reserved." They are, after all, rights. However, the phrase can potentially become significant when considering whether a party is estopped from asserting a claim that should have been raised earlier, or has made an unfavourable admission.