In addition to G. Smith's answer, it's worth noting the importance of the neutron-proton ratio.
Prior to Big Bang Nucleosynthesis (BBN) Two key nucleon reactions before BBN commences are:
- a neutron interacts with a positron to create a proton plus antineutrino (and vice versa), and
- a neutron interacts with a neutrino to create a proton plus electron (and vice versa).
Initially the temperature was high enough for all these reactions to take place, maintaining an equilibrium of protons and neutrons. However, as the temperature rapidly dropped, the neutron-proton inter-conversion rate per nucleon fell faster than the Hubble expansion rate, favouring protons ahead of neutrons. About one second after the BB, the temperature had dropped to about 0.7 MeV, too low for these reactions to continue (the "freeze-out"), by which point the N:P ratio had fallen to 1:6.
Nucleosynthesis begins
BBN was now spluttering into action, and the first multi-nucleon element formed is the simplest: deuterium (one proton plus one neutron). However, the temperature was still too high for deuterium to survive, as the energy of some photons was higher than deuterium's binding energy, and any $^2_1$H would quickly photodissociate. This period is called the "deuterium bottleneck"; but once the temperature had dropped to about o.1 MeV, the $^2_1$H could survive, and as a result there was a sudden and significant increase in the proportion of deuterium atoms present. And as helium-4 has the highest binding energy per nucleon among the lighter elements, almost all this deuterium quickly ended up as $^4_2$He.
However, free neutrons are unstable and decay with a half-life of 611 seconds, so any neutron that hadn't managed to get to "safety" within an atomic nucleus during this brief 20-minute BBN period was most likely to have decayed into a proton. As a result, the final N:P ratio ended up at 1:7.
N:P ratio predicts He/H ratio
This final 1:7 neutron-proton ratio tells us that for every 14 protons at the end of BBN, there were two neutrons. Since on average those two neutrons found themselves with two protons in a $^4_2$He nucleus, this leaves 12 protons without a partner. A proton on its own is a hydrogen ion, so we can predict that once the temperature had dropped too low for further nucleosynthesis (at around 30KeV, or about 20 minutes after the BB), there were roughly 12 hydrogen atoms for each atom of $^4_2$He.
[There were also trace amounts of other nucleosynthesis residues: about 0.01% of deuterium and $^3_2$He (helium-3), and one part in 10 billion of $^7_3$Li (lithium-7); and also even tinier amounts of the unstable isotopes $^3_1$H (tritium) and $^7_4$Be (beryllium-7), both of which soon decayed.]
So, this is what our knowledge of physics predicts:
By the end of Big Bang nucleosynthesis, on average 12 out of 13 atoms will be hydrogen (92%) and the remaining atom will be helium (8%); or by mass, with a helium atom four times heavier than a hydrogen atom, it's 4:12 or 4/16ths or 25% helium, and 75% hydrogen. And this is, indeed, exactly what we now find.
Even stronger evidence
While the question asks "How does the ratio of hydrogen to helium help prove the big bang theory?", there is even stronger support for the standard BBN model when we look at the lesser "relic" product of that brief burst of nucleosynthesis: deuterium. The production of this isotope is highly sensitive to the primordial baryon abundance, and observations of the Universe's large scale structure and of temperature fluctuations in the cosmic microwave background radiation have tightly constrained that number, enabling a well-bounded prediction for the relic abundance of D. This D/H prediction is in close agreement with the current abundance inferred from observations of low metallicity galaxies, providing even stronger support for the standard model.
Reference
For a more technical review of BBN predictions – including pre-BBN processes, N:P ratios and isotope abundances – see (for example) Primordial Nucleosynthesis in the Precision Cosmology Era (Gary Steigman, 2007).