saisir = to grab/seize/grasp
From what I understand, the use of "saisir" has the same flexibility as the English use of "to grab/seize/grasp" such as grabbing something physical or grasping something intangible such as a theoretical concept. Grasping a concept = understanding a concept.
In most situations, the French for "I understand your words" would be "Je comprends tes mots" (from the verb "comprendre"). "Je saisis" (from the verb "saisir") is most often translated as "I catch", either literally as in "I catch the ball" or figuratively as in the past tense expression "I did not catch what he said" meaning "I could not hear what he said". If you use "comprendre" as "to understand" and "saisir" as "to catch" you will not go wrong. Accept occasional confusion or meaningless statements in what I assume are computer generated sentences in duolingo. For more, see http://french.about.com/od/vocabulary/g/saisir.htm
I know that "saisis" can mean "to physically grab" or to understand a concept. I understood "mots" to mean "words" or "notes". I translated this as "I grab your notes" and was marked incorrect. If, for example, one student takes the handwritten notes (pieces of paper) of another student, how would I write that?