LTspice question using LT3625
The supercapacitors in that simulation start out very-nearly charged. In fact the charging ends just after 4ms.

If you change the SPICE Directive .tran 4m to .tran 200m then you will simulate for 200ms and be able to see exactly what is going on.

To start the supercapacitors discharged just change the directive .ic V(out)=4.4 to .ic V(out)=0.

They have given that simulation a very beefy solar panel which is why the charging current is high. You can replace that solar panel model with your own quite easily; there is an excellent example here: https://electronics.stackexchange.com/a/257426/193946