In plain English
Training teaches a model its capabilities. Inference-time compute is the processing power it uses when putting those capabilities to work. A simple question may require relatively little computation, while a difficult problem may benefit from extra reasoning, generating multiple possible answers, or checking its work. More computation can improve results, but also increases cost and response time.
Example
An AI solves a complex math problem by exploring several possible solutions and checking them before returning its answer, using more processing power than it would for a simple question.