The basic meaning of tanto/tanti is “in such a large quantity”.
In the simplest case this may simply conclude an explicit comparison. Take the famous lines by Dante:
dietro le venìa sì lunga tratta
di gente, ch'i' non averei creduto
che morte tanta n'avesse disfatta
In Dorothy L. Sayers's translation, this is “...there the folk forlorn / Rushed after it [a moving ensign], in such an endless train, / It never would have entered in my head / There were so many men whom death had slain”. So here tanta (feminine as it refers to gente) is rendered as “so many.” (This is alluded to in T.S. Eliot's The Waste Land: “A crowd flowed over London Bridge, so many, / I had not thought death had undone so many.”)
It is not necessary to make the comparison term explicit. If you said:
Non credevo che d'estate ci fossero tanti temporali.
you'd be implicitly saying “...as many as there actually are” or simply “so many”. To take one of the examples from Treccani's entry on tanto:
ci vuole tanto tempo per rispondere?
(“do you need such a long time to answer?”, more or less), here you are precisely referring to, say, the five minutes your interlocutor waited before answering.
Notice that in all of these examples, if you used molto/molti, either the sentence wouldn't work or the meaning would change. You cannot use molto in a correlation (Dante's sentence or one with an analogous structure wouldn't mean anything with molta). If you asked
ci vuole molto tempo per rispondere?
you'd just be asking whether an answer is due in a long time or not.
In common use, it's frequent to find tanto as almost a synonym for molto, but even then there is often a sense of correlation with something unexpressed or, at least, a particular emphasis. If I say
Ho tanta voglia di vederti
I'm saying “I have such a desire to see you”, as if implying that I can't stand it any more (“tanta voglia ... che verrei lì stanotte”, say). On the other hand “ho molta voglia di vederti” is just that my desire is, say, 8 in a 1-10 scale.
You may well find cases where tanto is used as a plain molto: complex or hyperbolic words and expressions are often attenuated through use.
Just think about troppo: its meaning is too much or too [adjective/adverb]. So, if I say that a thing is troppo bella, literally I'd be saying that it is “too beautiful” (so, for instance, we should find a less beautiful one for a particular use). But is it usual to hear this as a positive description, as if it meant that that thing is exceptionally beautiful.