Ah well, we simply have to work with what we're given, don't we? I think it's fairly reliable that this is, indeed, an insect: there seems a clear separation into a well-defined head, thorax, and legless abdomen, with no more than three legs visible on a single side of the thorax. The forward-protruding mandibles and antennae with broader basal segments are also insect-like rather than entognath-like (so it's not a stem dipluran or something like that). However, Garrouste et al. suggest that this specimen not only represents an insect, but also a crown-group pterygote. This I feel is a little more problematic.
Assignment of Strudiella to pterygotes relies on two characters: the relatively elongate legs, and the appearance of the mandibles. I suspect that it would not be that difficult for elongate legs to evolve convergently, and the supposed Carboniferous dipluran Testajapyx appears to have relatively long hind legs at least (Kukalova-Peck 1987). The mandible structure is a bit more difficult to hand-wave, though: Garrouste et al. interpret Strudiella as having an orthopteroid mandible, which is believed to be a synapomorphy of the Metapterygota, the particular clade within the pterygotes uniting the Neoptera and Odonata.
The earliest hexapods possessed a mandible with a single articulation (the condyle) to the head; such a mandible is still present in springtails, diplurans and bristletails. The clade uniting silverfish and pterygotes developed a second articulation (the acetabulum) on the inside of the mandible. In silverfish and mayflies, the acetabulum is anterior to the condyle, and the acetabular articulation is relatively loose. In the metapterygotes, the acetabulum has moved back to become more level with the condyle, and the mandible's articulation with the head is a lot more solid. The fossil remains of Strudiella do not appear to show the mandible articulation itself, but the general shape and orientation of the triangular mandible is more similar to the metapterygote arrangement than to the more basal morphology. Besides, such a morphology is has more clearly been demonstrated in the Lower Devonian Rhyniognatha, known only from a pair of preserved mandibles that are even older than Strudiella (Engel & Grimaldi 2004).
The ultimate question, then, is: is this one character enough to cement these taxa as crown pterygotes, with the implication that winged insects must have evolved considerably earlier than their fossil record currently indicates? Strudiella itself shows no sign of wings; Garrouste et al. suggest that it may be a nymph of a winged adult. I would counter that it also doesn't appear to possess any incipient wing buds, but of course it is debatable whether the preservation is good enough to be confident on this point.
If winged insects have been around since the Early Devonian, why do we find no direct evidence of them until the mid-Carboniferous? Wings are among the most commonly preserved insect remains—to the extent that if, as the adage goes, mammalian palaeontology is all about 'the tooth, the whole tooth, and nothing but the tooth', insect palaeontology often threatens to be 'all in vein'. For my part, I'm not inherently opposed to the idea of Devonian winged insects, but I don't think I'd really be willing to accept them until we're shown the actual wings.
Engel, M. S., & D. A. Grimaldi. 2004. New light shed on the oldest insect. Nature 427: 627-630.
Garrouste, R., G. Clément, P. Nel, M. S. Engel, P. Grandcolas, C. D’Haese, L. Lagebro, J. Denayer, P. Gueriau, P. Lafaite, S. Olive, C. Prestianni & A. Nel. 2012. A complete insect from the Late Devonian period. Nature 488: 82-85.
Kukalova-Peck, J. 1987. New Carboniferous Diplura, Monura, and Thysanura, the hexapod ground plan, and the role of thoracic side lobes in the origin of wings (Insecta). Canadian Journal of Zoology 65: 2327-2345.