First off, phonetics is not about features, though often in introductory classes if you don't have separate courses on phonetics and phonology, phonetics gets lumped together with phonology. Second, it is simply wrong to say that liquids and nasals are [+cons,-syll]. The misstatement comes from confusing the most common properties of a system with the inherent properties of the system.
Liquids and nasals are [+cons,+son] and if you want to distinguish syllabic versus non-syllabic liquids / nasals (the latter are more common), you can employ [syllabic] to do that – syllabic liquids/nasals are [+cons,+son,+syl] and nonsyllabic liquids are [+cons,+son,-syl]. That assumes the classical SPE feature system, and in fact syllabic was one of the first features to be disposed of in the 70's. Textbooks may use the SPE system because it does represent the last standard for representation, but there are many current theories of representation, and none of them employ a feature [syllabic]. There are many different accounts of the difference, virtually all of which rely on some prosodic object such as a mora, V-slot, or nuclearly prespecified X-slot (there is also a non-standard redefinition of [consonantal] where glides are [+consonantal] and corresponding vowels are [-consonantal], but as far as I know that has not been applied to the difference between syllabic and non-syllabic consonants.
It is a somewhat open question how "deeply" a syllabic / non-syllabic distinction is needed. There has always been a strong urge to get rid of the distinction, because most of the time you can predict where things are syllabic. In fact, if you also have syllables (which we don't always have), then you can always predict syllabicity. Problematic cases like the Swahili minimal pair mbuni "ostrich", m̩buni "coffee tree" become non-problematic if you include syllable structure – mbu.ni vs m̩.bu.ni. As it happens, the distinction in Swahili is more superficial and there is no need to admit an underlying contrast in syllabicity types for nasals – it can be predicted, once you have the rule system sorted out. For theories that posit moraicity as the bearer of the concept "syllabic", there isn't much hope of predicting "syllabicness" because moraicity is a fairly fundamental phonemic property (since it is also how you express long versus short).
There is no general answer to the question of how liquids and nasals become syllabic: there are many answers, and it just depends on the facts of the language. In English, it mostly has to do with unstressed syllables having əC where C is a sonorant. In many languages it has to do with extraneous consonants that can't be syllabified around a vowel but where there is a sonorant that can fill the bill (Sanskrit, for example). A number of languages of West Africa have CR̩V syllables with syllabic pre-vocalic consonants, and these often derive from disyllables (*pila) which lose a vowel (→ [pl̩a]).