What is the evidence for 'billions of neutrinos pass through your body every second'?

Those neutrinos come from the Sun. Fusion converts protons to neutrons, so that must produce neutrinos. One can calculate the number of nuclear reactions necessary for the power output, and get a number for the neutrino flux.

One can also estimated the flux from the cross section of the detector.

The two rates differ by a factor of about three. That was resolved by the neutrino oscillations between the three flavors (electron, muon and tauon neutrinos).

The existence of the neutrinos was established using energy and momentum conservation in neutron decays. There have been experiments with neutrino and antineutrino beams both at Cern and Brookhaven and have established their interaction crossection with matter. To get one neutrino interacting in the detector it means that thousands have passed without interacting, according to the theoretical calculations. Your "that we have never detected even a hundred neutrinos in a second through one detector? " is misleading, because the one we do detect, mathematically means that the calculated beam flux is correct according to the theory

There exists a solid theory that can estimate the number of neutrinos given certain assumptions of what the cosmic charged particle background is.

For example, we measure the muon flux at sea level, and muons decay into electrons and a muon neutrino and an electron antineutrino, so we know from the kinematics what the muon induced flux of neutrinos is at sea level. ( an average flux of about 1 muon per square centimeter per minute. far from billions)

There are detectors detecting solar neutrinos and those also agree with the mainstream theory of weak interactions. Those fulfill the billions recipe,

The flux of solar neutrinos at the earth's surface is on the order of $10^{11}$ per square centimeter per second.

Theory says that there should be cosmic relic neutrinos, coming from their decoupling in the Big Bang model, similar to the Cosmic Microwave Background, this would add orders of magnitude to very low energy background neutrinos, but this is still now a theoretical prediction.

As others have noted, the neutrinos come from the sun. Given that, there are two broad ways of estimating the flux of neutrinos: one is theoretical, and the other is experimental.

The theoretical way is based on the Standard Solar Model. This is a well understood model with solid experimental validation, and astronomers and astrophysicists therefore have great faith in it. According to this model, the solar neutrino flux is dominated by proton-proton fusion reactions, which generate an electron neutrino flux of approximately $6 \times 10^{10}\ \mathrm{cm^{-2}\ s^{-1}}$ at $1\ \mathrm{AU}$.

The experimental way is to build neutrino detectors and measure the flux. Because of the extremely small cross section, it is difficult to build a detector that can collect enough data to reduce the statistical and systematic uncertainty to qualify for a precision measurement. Nevertheless, a lot of work has been performed in this area and results have been obtained with uncertainties to within a percent or so.

The experimental results and theoretical predictions did not agree with each other; they were off by a factor of three, which was the so-called Solar Neutrino Problem. This was resolved by hypothesizing, and then experimentally verifying, that the electron neutrinos produced in the sun "oscillated" into other flavors of neutrinos (muon, tau) by the time they were detected on earth, so now the experimental neutrino flux measurements agree with the theoretical predictions.