"Rescue" in an experimental context means you're capable of undoing some experimental manipulation. It's considered evidence towards identifying a causal mechanism or verifying that your manipulation does what you think it did.
For example, let's say you think secretion of molecule ABC is key towards mounting an immune response to MouseKiller virus. You test this by creating a knockout mouse that lacks the enzyme that synthesizes ABC, let's called it Alphabet Synthase, and find out that all the mice die when exposed to MouseKiller virus.
However, maybe you got it wrong. Maybe ABC isn't the key, but actually Alphabet Synthase is involved in some other pathway independent of ABC. You do an experiment where you produce some ABC in the lab, inject it intravenously to your Alphabet Synthase-knockout mice, and find these mice no longer die when infected with MouseKiller. You can say that you've "rescued" the ability of the mice to survive MouseKiller, and now have better evidence that indeed ABC is the key factor.
Another example of a rescue experiment would be that you think specifically Alphabet Synthase in B-cells is important for defense against MouseKiller. It's not very easy to knock out Alphabet Synthase in B-cells specifically, but you can 1) knock out Alphabet Synthase entirely in some mice, and 2) introduce a new version of Alphabet Synthase specifically in B-cells (for example, using a Cre-dependent expression vector with a B-cell-specific promoter). Experiment (1) shows the knockout is susceptible to death from MouseKiller, experiment (2) shows you can rescue the immune defense if B-cells specifically express Alphabet Synthase.
The precise experimental manipulation described as a "rescue" is completely dependent on the research question being asked: you'll have to read specific papers to know what exactly they are doing as a "rescue".