It does mean "jumped", but you would always need a subject with it to express that meaning. "Léim mé": I jumped. The past tense (and not only) of the verb to jump just happens to end in -im. It's not a personal ending.
"Léim" on it its own could only mean "I read"; here, there is no extra subject pronoun, so it's clear that the -im means "mé"
(And indeed, you can say léann mé = "I read")
And the dictionary form is actually the imperative. This is the base form from which you remove the -igh and so on.
Examples of some dictionary forms/imperatives:
bí: be (present tense tá sé) léigh: read (present tense léann sé) ól: drink (present tense ólann sé)
and all those 11 irregular verb forms mentioned in the notes (abair, clois, tar, etc.)
This is actually a bit of an exception. Originally, the word would have been 'léighim', but I think it was shorted to 'léim' as a result of the spelling reform of An Caighdeán Oifigiúil (I might be wrong). The original spelling unambiguously identifies the root as 'léigh' and the declension suffix '-im'.
The vast majority of Irish verbs - except for this and perhaps some of the irregular verbs - are spelt unambiguously.
'clois' (to hear), 'cloisim' (I hear)
'ith' (to eat), 'ithim' (I eat)
'imir' (to play), 'imrím' (I play)