The is alluded to in the script but not completely spelled out. Immediately prior to seeing Eames at Fischer's office, we see him in a meeting with Cobb and Saito:
Eames: Can you get me access to this man here? Browning, Fischer Sr's right hand man. Fischer Jr's godfather.
Saito: It should be possible. If you can get the right references.
Eames: References are something of a specialty for me, Mr. Saito.
This implies that Saito uses his connections to get Eames a job directly with Fisher's organization or with some lawyers or consultants that would meet with Browning.
Eames's role as a 'forger' in the context of dream sharing is that he is an expert at impersonations and manipulating people. I think we are meant to presume that he uses those skills to get himself into a meeting where he can observe Browning directly - by pretending to be a lawyer or business expert.
This may seem unlikely, but backed by Saito he may have the resources to do this. All Saito has to do is to persuade or bribe someone to include Eames 'on their team' at a meeting. All we see Eames do is sit, observe and take notes.