Mechanism of homologation of aldehyde to alkyne: Ohira–Bestmann reaction
This reaction is the Ohira–Bestmann modification1,2 of the Seyferth–Gilbert homologation.3,4
In the original Seyferth–Gilbert homologation, a diazomethane phosphonate is used to convert carbonyl compounds to alkynes with one extra carbon in the presence of potassium t-butoxide. I used acetophenone as the reactant here just to simplify things, but the same mechanism applies to your substrate.

In the last step, the diazoalkene may be equivalently thought of as a carbene. You could lose nitrogen to form a carbene, which undergoes a 1,2-alkyl shift to generate the alkyne. I have drawn it as being concerted but I am not entirely sure whether it is really concerted (as far as I know, there is still some degree of contention over the Wolff rearrangement — but feel free to correct me if I am wrong).
This procedure has some issues due to the strongly basic conditions. For example, in your substrate, the presence of t-butoxide can easily lead to racemisation α to the carbonyl group. The Ohira–Bestmann conditions are much milder and generally do not lead to racemisation.
As Waylander mentioned in a comment, you need to specify the solvent, but generally the Ohira–Bestmann version is carried out with potassium carbonate in methanol. The combination of carbonate and alcohol generates small amounts of alkoxide, which then attacks the carbonyl group in the diazophosphonate to produce the same reactive species:

However, I looked up your compound on Reaxys and it seems that the authors used caesium carbonate in t-butanol.5 I am very hesitant to suggest t-butoxide attacking a carbonyl group as a nucleophile, so I would go out on a limb and propose an alternative mechanism. The ketene generated can probably be intercepted by the t-butanol to generate t-butyl acetate. I don't have any evidence for this, however.

In general you had a lot of good ideas (Wittig-type chemistry and a Wolff rearrangement) and were mostly on the right track. The missing ingredient was probably the solvent, which is crucial for the generation of the reactive carbanion.
References

Ohira, S. Methanolysis of Dimethyl (1-Diazo-2-oxopropyl) Phosphonate: Generation of Dimethyl (Diazomethyl) Phosphonate and Reaction with Carbonyl Compounds. Synth. Commun. 1989, 19 (3–4), 561–564. DOI: 10.1080/00397918908050700.

Müller, S.; Liepold, B.; Roth, G. J.; Bestmann, H. J. An Improved One-pot Procedure for the Synthesis of Alkynes from Aldehydes. Synlett 1996, No. 6, 521–522. DOI: 10.1055/s-1996-5474.

Gilbert, J. C.; Weerasooriya, U. Elaboration of Aldehydes and Ketones to Alkynes: Improved Methodology. J. Org. Chem. 1979, 44 (26), 4997–4998. DOI: 10.1021/jo00394a063.

Kürti, L.; Czakó, B. Strategic Applications of Named Reactions in Organic Synthesis; Elsevier: Amsterdam, 2005; pp 402–403.

Brenneman, J. B.; Martin, S. F. Application of Intramolecular Enyne Metathesis to the Synthesis of Aza[4.2.1]bicyclics:  Enantiospecific Total Synthesis of (+)-Anatoxin-a. Org. Lett. 2004, 6 (8), 1329–1331. DOI: 10.1021/ol049631e.