Mechanism of rearrangement reaction found in synthesis of retinol
When trying to propose a mechanism, especially one where the bond changes are not obvious, I like to start by numbering the atoms of the product and trying to map them to the reactants. Here, I've taken the numbering given by ChemDraw for the product and translated it to the reactant.



Carbons 1-3 are pretty logical. In the product, carbon 4 is alkyl with 2 hydrogens and between to other carbons. In the starting material, the carbon connected to C3 is a carbonyl with zero hydrogens. It seems unlikely to me that this would be C4. 

So I look for another, more logical mapping. For example, carbons 7 and 9 are methyl groups in the product, and there are two methyl groups in the starting material, which I label as 7 and 9. In both structures, they are bridged by a single carbon, which I call 6.

In the product C5, is part of an alkene and so is the carbon connected to C6 in the reactant, so that seems logical. And the next carbon in the starting material has two hydrogens, so that seems a better mapping as C4. The last carbon of the reactant must be lost as carbon dioxide, so I've labeled it C10.

With a hypothetical atom mapping, now I can identify bonds broken and formed. C10 and its adjacent oxygens are being lost as carbon dioxide, which makes it easy to identify the bonds broken as C3-C10 and C6-O, with a shift of the pi-bond from C4-C5 to C5-C6. 


Without being aware of [3,3]-sigmatropic rearrangements, or the Claisen rearrangement specifically, you might be stuck. The Claisen rearrangement is a cyclic movement of electrons converting an allyl vinyl ether into a gamma,delta-unsaturated carbonyl compound. This is driven by the formation of a carbonyl at the expense of an alkene.



The given starting material isn't quite an allyl vinyl ether, but as a 1,3-dicarbonyl, it is equilibrium with one by tautomerization. From the tautomer shown, the Claisen rearrangement is possible, which forms the C3-C4 bond we noted earlier and shifts the pi bond into the desired position. Also note that the resulting intermediate is the one shown in the solution manual. Decarboxylation through a similar cyclic movement of electrons gives the tautomer of the product along with carbon dioxide.