I am looking to break a lease with my landlord and am looking to hire an attorney.
What does "Client assigns to counsel all rights conferred by statute or rule to recover attorneys' fees and costs from this lease termination" mean?
What does this mean in this engagement letter?
The statement means that you will not oppose your lawyer's legal entitlement to (1) reimbursement of litigation costs he incurred and (2) payment (i.e., fees) for his services in the event that the lease is terminated. Since the clause is in terms of "assigns", it purportedly refers to recovering from the adversary those fees and costs. It does not preclude recovery from you, though. The term "costs" refers to money a litigant has to pay for actions such as filing a motion in court, mailing documents to the adversary, obtaining deposition & court hearing transcripts, and so forth.
This clause altogether is so redundant and so sloppy I would hesitate retaining a lawyer with that "quality". For instance, in general the default presumption is that nobody performs services for free. Absent any indication in your matter that contradicts or supersedes that default presumption, there was no need for this lawyer to include the statement you quote. The lawyer is getting paid either way.
Note that the statement refers to the outcome of lease termination (i.e., contingent to you prevailing in the legal dispute). Except for very specific scenarios, no prevailing litigant would volunteer to cover fees which are legally recoverable from his adversary. If anything, it would be easier for a lawyer to bill his client than going to court again to litigate against the adversary the matter of attorney fees. Thus, the statement is tantamount to saying "you the client will allow me to take the longer path as to attorney fees if we win the underlying case".
It is also pointless for the clause to have the exact same statement at the beginning and at the end of that paragraph. It reflects that also this lawyer is in the business of doing copy/paste carelessly.