How do foam core panels help these problems?

First, MEPS has an R-value of 4.35 per inch of thickness at 25°F. The typical panel we use has 5-5/8 inches of foam in its core (R-26); this is the composite panel calculated R-value at 25° F.
Second, the panels are manufactured in large sheets (4'X8' up to 8'X24') consisting mostly of insulation. The seams are sealed, creating continuous blanket insulation uninterrupted by structural framing members (as opposed to a conventional stud wall with a break in the insulation ever 16 inches to accommodate framing lumber).
Removing the framing lumber from the insulation allows full utilization of the MEPS. Remember, wood is relatively poor insulator (R-1 per inch of thickness), and in a wall with studs every 16 inches, wood can account for up to 30 percent of a wall's volume, significantly reducing the overall performance of the wall. The stated R-value of an panel is a true reflection of its ability to resist conductive heat loss.
Third, in panels the continuous mass of high density foam nearly eliminates the air movement within the wall and the heat loss caused by convective looping of air within the insulation. Compare the continuous mass of MEPS to a low density insulator such as fiberglass. Fiberglass is the material used to make filters for face masks, furnaces and cigarettes because it allows air to pass through.
Fourth, MEPS foam is only marginally affected by moisture condensation. In one test, MEPS was force saturated to ten times its normal dry weight and still retained 80% of its insulating value. In addition, MEPS insulation is inert, organic material. It provides no food value to plants, animals and microorganisms. It will not rot and it is highly resistant to mildew.