The words nō, natō, adnō are all verbs which means to swim. They have their own conjugation respectively.
Maybe there are not only three, but I'm not sure.
I wonder why there are several words in the same meaning and are formed from the same root word?
The word nato is derived from no, so they ultimately have the same root. The verbal suffix -t(a)-o was attached to the stem na- to form na-ta-o (contracted to nato). This suffix usually has an intensifying or repetitive function, so the the original meaning of nato might have been "to float about, be tossed about" or similar; but often the suffix has lost most of its meaning, so it may be used to mean just "swim".
It is very common to use this suffix to create new verbs. The -t- is basically the same suffix that forms supine stems and hence past participles (so it can also be something other than -t-, such as -s-). And the same suffix can even be stacked:
There are two basic forms of the verb: natare and nare (or nato and no if you prefer first person singular to the infinitive). This amount of variation is not unusual.
A number of other swimming verbs are obtained with prefixes you can add to either one: ad-, e-, de-, in-, prae-, re-, tra-, ab-. As with many verbs of motion, you can give details of the motion with a prefix. Are you swimming back, in front of someone, across something, away from something, to something, in a group, down the stream, or something else? Just add a prefix.
The meanings are not exactly the same, although all of them can be translated in many contexts as a mere "swim".
As pointed out by Joonas, in Latin "a number of other swimming verbs are obtained with prefixes you can add to either one: ad-, e-, de-, in-, prae-, re-, tra-, ab-".
As a case study to show the importance of separating genetic from typological classification of languages, I think that the present question becomes even more interesting if (re)formulated in the following terms: why is it the case that Latin, compared to Romance languages, has many complex/prefixed verbs for swimming (e.g., abnato 'to swim away', eno/enato 'to swim out', praenato 'to swim by', reno 'to swim back', subnato 'to swim under', trano 'to swim over', etc.)? To put it in the following typological (not genetic!) terms, why is it the case that, concerning this aspect of word formation, Latin is quite different from Romance languages and, in contrast, is very similar to Slavic languages (where the path/directionality is also typically expressed by a prefix)? So the answer to the interesting question Why there are several words for swimming in Latin is essentially the same to the question Why there are several words for swimming in Slavic. Is there any native speaker of a Slavic language to confirm this typological point?
For those readers interested in knowing more about the previous important typological change from Latin to Romance languages (for example, if you want to know why Joonas's list of Latin verbs above disappeared or is not productive in Romance), I recommend you the reading of the following book: https://benjamins.com/catalog/cilt.331 .
Finally, as for the interesting typological parallelism between Latin and Slavic, take a look at https://global.oup.com/academic/product/the-morphosyntax-of-transitions-9780198733287?cc=es&lang=en&# (You'll see that both Latin and Slavic belong to the SAME typological class: so-called "weak satellite-framed languages").
