If you haven’t already, you would find it helpful to look at the LSE Offer Pack for Undergraduates or Graduates.
A conditional offer refers to:
‘You do not yet satisfy the academic conditions required before entry but you plan to do so. Your conditions are given in the offer letter and may include tests such as English language as well as degree results, or attendance at Pre-sessional/ Introductory or LSE Summer School courses.’
In the event that you failed to meet one of the offer conditions (i.e. your degree result was not what you expected) you should still send your results and the Selectors will consider whether your offer can be confirmed.