Most French words are derived from Latin roots, as are Spanish words. So the languages have a lot of cognates. Of course, words sometimes evolve differently, so having the same etymology doesn't guarantee having the same meaning. But it helps.
Many families of French words come into two subfamilies: popular derivations (formes populaires), which have drifted quite a bit since Latin, and scholarly derivations (formes savantes), which were reinjected during the Middle-Ages and are closer to the original. Typically, the main, everyday noun has come through popular derivation while more “highbrow” terms such as adjectives and rarer nouns have come through scholarly derivations. Scholarly derivations can often be transposed into Spanish (and other Romance languages) by simple transformations. Occasionally the basic French word has a different origin but other words in the same cognitive area are close to the Latin root.
Here are a few examples.
- Fr. main (hand); Fr. manuel (by hand); Lat. manus; Sp. mano.
- Fr. arbre (tree); Fr. arborescence (tree shape); Lat. arbor; Sp. árbol.
- Fr. eau (water); Fr. aquatique (of/from/related to water); Lat. aqua; Sp. agua.
- Fr. cheval (horse); Fr. cavalier (horse rider); late Lat. caballus; Sp. caballo.
- Fr. roi (king), royal (royal, the “everyday” word); Fr. régalien (royal, in a legal context, e.g. a regal prerogative = prérogative régalienne), régir (to rule, as in to determine an outcome); Lat. rex, regis; Sp. rey.
- Fr. mère, père (mother, father); Fr. maternel, paternel (motherly, fatherly); Lat. mater, pater; Sp. madre, padre.
- Fr. bouche (mouth); Fr. buccal (relating to the mouth); Lat. bucca; Sp. boca.
- Fr. langue (tongue, language); Fr. linguistique (linguistics); Lat. lingua; Sp. lengua.
- Fr. milieu (middle, milieu); Fr. médian (which is in the middle) (different origin); Lat. medius; Sp. medio.
A similar phenomenon happened with many French irregular verbs, where the infinitive and the present indicative have drifted a lot but other tenses or derivatives are closer to the original and closer to the Spanish word.
- Fr. dire (say); Fr. dicible (which can be said), diction (diction); Lat. dicere, dictum; Sp. decir.
- Fr. entendre (hear); Fr. ouïe (hearing) (this one is a completely different root); Lat. intendere vs. audire; Sp. oír.
- Fr. aller (go); Fr. j'irai (I will go — here the French verb is very irregular with three different roots for the different tenses); Lat. ire, itum; Sp. ir.
- Fr. pouvoir (can, be able); Fr. possible (possible), potentiel (potential); Lat. posse, potui, possum; Sp. poder.