The claim that there are no syllables is based on the lack of evidence that the syllable is necessary, so this is an Occam’s Razor argument. If no language presents sufficient evidence that syllables exist, we cannot legitimately say that syllables exist, and that is the claim (i.e. that no language has been shown to have such evidence). However, all current theories have things that do part of what a syllable does.
It would help to say what a “syllable” is in linguistic theory: it is a grouping of some number of segments. That is about all one can say without getting into matters of controversy, or definition by exemplification. Syllables have one peak, though not all theories of syllables actually notate something as a syllable peak. A sequence like [lado] will not be parsed as one syllable, it will be two syllables, but [lao] might be one syllable or two (whence the need to insert syllable boundaries on occasion).
No phonetic definition or diagnostic of syllable has been uncovered, and John Ohala is a well-known phonetician who is skeptical about the syllable.
However, he does not deny that there are important modulations in the speech stream. Phonologists have a certain disinclination to embrace entities which cannot be clearly diagnosed in terms of the speech signal, thus we believe in segments because we can hear them and they can contrast, and we don’t easily believe in syllables because we can’t hear them as such and they don’t contrast.
Focusing on phonological considerations, what persuades most phonologists of the existence of the syllable is its utility for explaining phonological facts. One such fact is the pattern of consonant co-occurrence in many / most languages. In English, words can start with s+C or C+L (L= r,l) but not two stops, e.g. [*pka], and not a liquid plus stop (*[lpa]). This is not just a fact about word beginnings, it’s a fact about syllables, the standard analysis says: syllables cannot begin with [*pk, *lp]. Words (and syllables) also can’t end with [*pk, *kp] nor with [*pl]. (I will comment on the question of whether English has any syllables ending in pl below). You can’t just say that the phoneme sequences lp, pk, pl are blocked (help, napkin, applaud), but if you put an obstruent before lp or pk, you have an impossible sequence. The explanation is that obstruent+[l] or obstruent+[p] can’t go at the end of a syllable, and lp cannot go at the beginning of a syllable, so there is no possible parsing of obstruent+lp, obstruent+pk into syllables. Thus the syllable has utility is explaining this kind of gap.
It is uncontroversial that English does not have surface word-final [Cl] apart from [rl], but there are syllabic liquids, so [æpl̩] is a word. The existence of [æpl̩] does not automatically invalidate the segment-sequence argument for the syllable, especially since syllabic sonorants derive from əC anyhow (thus [æpl̩]←/æpəl/). It is possible to derive [æpl̩] from /æpl/, where final sonorants are forced into a syllable regardless of the preceding consonant – and then they are made syllabic. However, this ploy would not be available for *nakpkin.
The problem is that this sequencing argument doesn’t necessarily diagnose the syllable, it really diagnoses the onset and the coda (an onset cannot contain lp or pk, a coda cannot contain pk,kp). If we have the structural units “onset” and “coda”, is the syllable really necessary as well? Hyman in his book A theory of phonological weight expresses skepticism regarding the syllable on the grounds of non-necessity, in light of other phonological units that are motivated, in his case, the weight unit (now known as the mora), and he has an account of syllabicity without having syllables.
The most powerful argument for the full power of the syllable that I know of is the constituency argument set forth by Clements & Sezer, that in Turkish, a velar palatalizes if it is in the same syllable with a front vowel. You get palatalized k in [nekʲ.tar, ʃykʲ.ran, fa.kʲir, kʲir, dikʲ, kʲyrkʲ], and not in [uk.te, ak.rep, ok, i:.kaz, fark, zamk, bok.sit], where in the latter contexts, the velar is not in the same syllable (as onset or coda) as a front vowel. The syllable cannot be avoided by requiring the target and trigger to be adjacent (see kʲyrkʲ).
I’m a bit puzzled at why this argument hasn’t been cited universally as the knock-down argument that Turkish, at least, must have syllables, which makes me wonder if there is a factual issue. It is true that [kʲ] is a marginal phoneme (there are pairs like [kar] "snow", [kʲar] "profit", but generally [kʲ] is predictable. One could legitimately dispose of [kʲyrkʲ] by positing that it is underlying /kʲyrkʲ/ or /kyrkʲ/. However, it is widely assumed (gratuitously, in my opinion) that predictable feature specifications should not be included in underlying forms, so positing /kyrkʲ/ is an unlikely alternative analysis in the theoretical milieu where the Clements & Sezer analysis was posited. So I don't think we can explain the silence over Turkish based on thinking that there is a self-evident alternative that avoids mentioning the syllable.
I leave untouched the theoretical question of whether we can posit that all languages have syllables if it can be shown that at least one language has syllables. The anti-syllable claim is, simply put, that no language has been shown to require syllables, as well as requiring smaller multi-segment groupings.