The simpleton summary of Malthus's thesis is that population rises geometrically, but food supplies only arithmetically, and hence no matter what we do, there will come a time of starvation.
I remember this fact being presented to me in grade school. I was impressed with its profundity: population keeps doubling, but food only adds 1 at a time.
In fact, this version of Malthus is idiotic but true. It is also true that if my weight doubles every year and you only add a pound each year, then (as long as my initial weight is positive) I will eventually outweigh you. And it is also true that that's not how our weights increase over time.
Why aren't we all dead of starvation? Population certainly has the potential to increase exponentially, but this behavior is not observed long term, and it does not halt because of food shortages. Food production seems to keep up and even advance ahead of population growth, and there is, in particular, no logical or physical reason to suppose that food production can only increase arithmetically.
The simpleton version of Malthus's thesis can, however, be blamed on Malthus himself and his choice of an example:
If the subsistence for man that the earth affords was to be increased every twenty-five years by a quantity equal to what the whole world at present produces, this would allow the power of production in the earth to be absolutely unlimited, and its ratio of increase much greater than we can conceive that any possible exertions of mankind could make it ... yet still the power of population being a power of a superior order, the increase of the human species can only be kept commensurate to the increase of the means of subsistence by the constant operation of the strong law of necessity acting as a check upon the greater power.
To summarize, Malthus sets up a straw man argument: suppose food production increased by a huge amount, but only arithmetically, and suppose population increased exponentially. Why, then, eventually, population would exceed the necessary food.
Well yes it would, and it doesn't.