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A Poisson model estimates how many goals each team is likely to score, then uses the Poisson distribution to turn those two averages into a probability for every possible scoreline. Add up the scorelines where the home team wins and you have a home-win probability; add up those with three or more goals and you have an over 2.5 probability. It is the standard statistical starting point for football modelling because goals are relatively rare, roughly independent events - which is the situation the Poisson distribution describes.
The inputs are each team's recent scoring and conceding rates, usually adjusted for home advantage and for the strength of the opposition faced. The output is not a prediction of one scoreline but a probability distribution across all of them.
Its weaknesses are well known and worth stating. It assumes the two teams' goals are independent, which they are not - a team that goes 2-0 down changes how it plays. It does not see injuries, motivation, rotation or weather unless you feed those in separately. And it is only as good as the data behind it; thin or stale form data produces confident nonsense. A Poisson number is a baseline to compare against the market price, not an oracle.
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