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Thomas Robert Malthus is a 18th century demographer and economist that is famous for the study of population growth, especially for the support of laws that would limit the growth of (poor) population. The basis of his reasoning is summarized by Wikipedia:

The main point of his essay was that population multiplies geometrically and food arithmetically, therefore whenever the food supply increases, population will rapidly grow to eliminate the abundance.

On which basis are these two claims in bold funded?

The fact that a population grows geometrically (in other words, exponentially) in a situation of abundance of resources is now classic and known as the exponential growth or natural growth. This Wikipedia page claims that the model was in fact introduced by Malthus himself.

Why did Malthus consider that resources grow arithmetically?

Actually, the same Wikipedia page quotes Malthus (emphasis added):

"Through the animal and vegetable kingdoms, nature has scattered the seeds of life abroad with the most profuse and liberal hand. ... The germs of existence contained in this spot of earth, with ample food, and ample room to expand in, would fill millions of worlds in the course of a few thousand years. Necessity, that imperious all pervading law of nature, restrains them within the prescribed bounds."

The text does not mention arithmetic but bounded resources. It is more in line with a logistic model with a constant carry capacity, or seasonal variations.

Did Malthus really consider resources growing arithmetically, or is it a later misattribution?

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As mentioned in the question the exponential population growth was argued to exist because it is based on the 'natural growth' that can be empirically observed anywhere in nature where the resources are abundant. So you already answered that part pretty much yourself.

When it comes to the arithmetic growth of resources, the word resource is actually not appropriate modern economist would call things like food and shelter output not a resources which are factors of production (the wikipedia in this article uses the word resources improperly see Samuelson & Nordhaus Economics). Resources are inputs not outputs of production.

The reason why Malthus thought that output/production would grow arithmetically is due to diminishing marginal product of factor of production. Diminishing marginal product means that although output increases when you use more factors (resources like land, labor, capital) the output increases at a diminishing rate. The diminishing returns as argued by many economist must apply otherwise even single plot of land could produce enough crops to feed whole world. At some point just throwing more labor and capital at a plot of land will simply lead to smaller output.

Malthus thus reasoned that due to these diminishing returns output will over time grow only arithmetically as given the fact that amount of land is fixed in the world at some point no matter how much machinery and labor force you apply to it at some point the output will increase only minimally (e.g. having 3 tractors for one acre field might increase output significantly compared to having 1 but having 100 tractors as opposed to 97 will not make much difference).

However, Malthus turned out to be wrong because he did not taken into account the effect of technology on production. A Malthusian production function would look something like:

$$Q=K^{0.3}L^{0.2}$$

which exhibits diminishing returns to scale as more and more inputs of labor and capital (resources) are put in the output will always increase but at diminishing rate (in fact if you would plot this curve you would see that as factor inputs increase the output increase becomes more and more ‘flat’ and ‘linear-like’ e.g. imagine a plot of square root function).

However, modern economists realize that technology plays a role in production process too. A modern production function would look like:

$$Q=AK^{0.3}L^{0.2}$$

where $A$ would be the level of avaiable technology. So as long as technology keeps accumulating and growing sufficiently fast enough to keep output with population growth Malthusian trap will not occur.

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    $\begingroup$ Great answer but minor point. You haven't specified anything about the growth rate of A. The growth rate of A must be fairly fast, especially once the world is developed and K and L are already large. It seems to be fairly large, (since we haven't had the catastrophe yet), but who knows what the future holds? $\endgroup$ – RegressForward Oct 11 '20 at 23:05
  • $\begingroup$ @RegressForward that is a good point. Since the answer wanted to just know about Malthus theory I considered the technology part a tangent and did not go deep in it but you are of course completely right. This being said it seems that population also no longer grows geometrically especially in developed countries. $\endgroup$ – 1muflon1 Oct 11 '20 at 23:12
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Malthus says he thinks it grows arithmetically. One has to believe that Malthus knows what he thinks!

The "seeds of the earth" quotation you cite here is not meant to talk about food (despite the use of the word seed). Rather, that quote is to talk about the rapid growth rate of living things more generally, of which humans are a part.

The line above it in the wikipedia page helps disambiguate (emph. added):

Malthus wrote that all life forms, including humans, have a propensity to exponential population growth when resources are abundant but that actual growth is limited by available resources:

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