I think you're going to need an electric service upgrade if you want to use this water heater.
Your 100 amp service is already somewhat loaded, as well as the box being physically full.
If you call for hot water while the dryer or range is in use (as obvious heavy loads on your panel), you may well trip the main, if you do not upgrade to 200 amp service before installing this water heater.
A different water heater might make more sense, unless you have other reasons to upgrade your electric service; one of the heat-pump water-heaters (or a heat-pump-add-on to an existing water heater tank), perhaps.
Edit, Add: Unless this is a house that stands vacant for long periods of time, there's surprisingly little "standby loss" (what you save by using an instant tankless electric resistance heater .vs. a tank-type electric resistance heater that's insulated to modern standards) - and if it does stand vacant for long periods, you can switch the water heater off when you leave. Changing to a heat pump water heater (HPWH) can, by contrast, save quite a bit and get you "free dehumidificaton" in the summertime. If you have gas available, it's more of a toss-up - gas and HPWH can be similar in cost to run, and a gas water heater will usually cost considerably less to buy. Most of the cost of running a normal (modern, well-insulated) tank-type water heater is, in fact, the cost of heating the water that you actually use. Most households will see little savings from using an instant-tankless waterheater, in practice.
I find it a little puzzling that you have a modern panel inside your home and a separate fuse box dedicated to the dryer.
Confirm that this is the case - double check your modern panel to see if you've overlooked a 220v circuit that has tripped.
If it is a simple tripped circuit, reset it. If it happens again call an electrician to find out why.
Otherwise:
Best Answer
Assuming that there are no service capacity constraints, you have a few options:
Replace some of the breakers with tandem breakers. These essentially double the density of each slot. These breakers come in a variety of configurations of both 120/240V and combos with both. This is the easiest of options for a do-it-yourselfer to complete.
Install a sub-panel. You would relocate a number of circuits from the current box to a new sub-panel.
Replace the entire breaker panel with a larger panel. This is the most difficult of the three options and sometimes requires the utility to disconnect your service in order to complete this.
As always, contact a licensed electrician if you are unsure of how to safely make these modifications.