How does one prove they meet these two tests: the bona fide residence test and the physical presence test to claim foreign earned income?

Physical Presence Test. You were present in foreign country for full 330 days in any period of 12 months in a row. (That is these 12 months can include two tax years.)

Bona Fide Residence Test. If you are citizen of the foreign country you meet this test, unless you were in the foreign country for definite temporary purpose.
No specific rule determines if you are a bona fide resident of a foreign country because the determination involves your intention about the length and nature of your stay. Evidence of your intention may be your words and acts. If these conflict your acts carry more weight than your words.
Generally, if you go to a foreign country for a definite, temporary purpose and return to the United States after you accomplish it, you are not a bona fide resident of the foreign country. If accomplishing the purpose requires an extended, indefinite stay, and you make your home in the foreign country, you may be a bona fide resident.

First, download a copy of Publication 54.
Publication 54 (2008), Tax Guide for U.S. Citizens and Resident Aliens Abroad
Where these two tests are explained in great detail.

To be a BFR you have to be a US citizen or a US resident who is also a citizen of a country with whom the US has a tax treaty

Then you have to be in that foreign country for an entire tax year (Jan. 1-Dec. 31.)

Then it is a matter of facts and circumstances; you have a home there, a drivers license, a residency card of some kind, bank account, church and/or club affiliations, you pay tax there if tax is due, etc. Things that indicate that this is just not a fleeting visit