This phrase basically means “Please, allow me to help you!”
Mike Brockington has some good examples of situations where this phrase can be used. In general you can use it whenever you see someone who looks like they’re struggling with something and you think you can help solve the problem. The problem could involve manual labor (like struggling to carry something heavy, or trying to maneuver a door while carrying a lot of things, as in Mike’s examples).
You could also say this to someone who is having a mental struggle rather than a physical one. For example, maybe you see a person standing in front of 10 different recycling bins, and they are holding an empty soda can, unsure which bin it should go into. If you are an expert on this collection of recycling bins you could reach out your hand and say “Please, allow me!” - which would tell the person that you want to take their soda can and recycle it for them since you know where it belongs.