There were several "moles" in the Manhattan Project — scientists who were working for the US/UK but were also secretly working for the USSR. They gave the USSR many pieces of information, primarily the results of scientific experiments and design work on the atomic bomb itself. The USSR used this information as a "check" and "guide" on the work of their own scientists, who were largely unaware there was espionage information at all. The conservatism of the head of the Soviet project led them to basically copy the Nagasaki device, because he wanted to make sure they could do something that was known to work first before innovating further, even though the Soviet weapon designers figured out better ways to design the bomb.

If you are interested in the kind of information passed, this blog post has many declassified drawings from the Soviet archives that describe how the Nagasaki bomb worked. The main Los Alamos "moles" were Klaus Fuchs (German physicist working for the UK; the highest-ranked and most-talented of the "moles"), Theodore Hall (a very young American physicist), and David Greenglass (an American engineer, the brother-in-law of Julius Rosenberg). There were a few spies at other sites as well (George Kovals at Oak Ridge, for example).

Designing the bomb is only one part, not even the most difficult part, of making an atomic bomb. The hardest part of the job, for both the USA and USSR, was refining the fuel for the bombs (enriched uranium or separated plutonium). The espionage may have helped them in some areas, but most of the work they necessarily had to do on their own. So it can be over-emphasized.