Fluorescein synthesis
This reaction is considered a modified Friedel-crafts acylation. The reaction is formally a series of electrophilic aromatic substitutions, with hydroxyl leaving groups picking up protons (1), however the Lewis acid catalysis obviously plays a role in the mechanism. I think it increases the electrophilicity of the phthalic anhydride in the first step, and facilitates the elimination of the hydroxyl groups. For the mechanism without the involvement of the zinc(II) chloride, see (1). Here is a mechanism emphasizing the role the Lewis acid might play in catalyzing the reaction:

This reaction works in a similar way to the Nencki reaction (2)
In order to get the form of the molecule shown as the product of your reaction, the $\mathrm{pH}$ needs to be increased in order to open the lactone ring.(3) This is fluorescein-A, the form of the molecule that actually fluoresces.

References:

Jag Mohan, In Organic Analytical Chemistry: Theory and Practice; Alpha Science International Ltd.: Pangbourne, England, 2003, p. 497 (ISBN: 1-84265-119-6).
M. Nencki, N. Sieber, &quot;Ueber die Verbindungen der ein‐ und zweibasischen Fettsäuren mit Phenolen,&quot; Journal für Praktische Chemie 1881, 23(1), 147-156 (https://doi.org/10.1002/prac.18810230111).
Clifton J. Stephenson, Ken D. Shimizu, &quot;A fluorescent diastereoselective molecular sensor for 1,2-aminoalcohols based on the rhodamine B lactone–zwitterion equilibrium,&quot; Org. Biomol. Chem 2010, 8(5), 1027-1032 (https://doi.org/10.1039/B918823A).