"We are eating."
"Nous sommes mangeons" would add 2 conjugated verbs, which is not correct (we are eat, he is eats...).
"Are eating" is a continuous present formed with the auxiliary "to be", conjugated, and the present participle/gerund form of the verb (-ing)
Such a construction does not exist in French.
"We are eating" translates to "nous mangeons", and "we eat" as well.
There is no "avons" (= have) in this sentence.
"We are eating" describes what we are doing at this very moment, but the continuous present tense does not exist in French. As a consequence, both "we are eating" and "we eat" must be translated to the French present tense: "Nous mangeons."
« French » is an adjective describing the subject « we ». The translation is direct: « Nous sommes français(e)s ».
« Eating » is a verb form (present participle) and « are eating » describes an on-going action. This verb form is the continuous present tense.
The continuous present tense does not exist in French and there is no direct translation : Sommes mangeant (present participle) does not mean anything.
« To be + verb-ing » must translate to the French present tense.
« We are eating » = nous mangeons.