As a town that's not really on the international tourist trail, the food options in Kampaeng Phet are a little thin, especially where tourist-friendly establishments are concerned. While, like any Thai city, there are lots of roadside food stalls, there are very few restaurants around town, and many do not have English menus.
The one area of town where you will find a lot of restaurants is along the west bank of the river, which is technically the town of Nakorn Chum. These restaurants are definitely Thai style eateries - ramshackle buildings right on the river's edge, open-air for the most part and built to accommodate large groups. They are definitely worth a try for at least one meal, but be warned that some do not have English menus.
The other big food area, for dinner at least, is the large night market near the riverfront in the new city. A large portion of the market is devoted to food, either to take home or eat there. You'll find just about every kind of Thai and Chinese street food you can imagine. These places are for the adventurous, but they are generally quite good, according to all reports.
One last area to look for food is the alley one block southeast of Kampaeng Phet Road, near where it dead-ends at the clock tower and provincial hall. The alley is lined with food stalls and seems to go all day, although some places are only open for lunch or dinner. The stalls seem to mainly serve the workers at the provincial hall, but there are some good eats here.
As for 'proper' restaurants, I can make only a few recommendations for places outside of hotels. M&P The Corner is a small café in a flatiron shop-house where Tesa 1 Soi 13 peels off from Ratchadamnoen 1 Road near the Riverview Hotel. The restaurant features a basic menu of Thai favorites and is a good place for a simple meal. Very nearby, in the base of the Riverview Hotel building (but apparently not associated with it) is the oddly named I'm Bakery & Beverage, a little coffee shop that has a nice display case of cakes, and also serves tea.
Over on Kampaeng Phet Road near the traffic circle is Coffee Today, perhaps the closest thing you'll find to a US style coffee shop. It has the usual suspects of coffees as well as a nice case full of pastries. I ended up here a couple of times, as it is the closest place to the historical park where I could cool off and stay dry (as it was the rainy season). The coffee is good and there's free wi-fi, when it works. One more coffee place worth a mention is a little wooden shack on Tesa 1 Road called Kampaengphet Ancient Coffee, although the sign is only in Thai. They serve a very traditional style of coffee, with sweetened milk. Order it iced and it will be served in plastic bag blown up like a balloon, put in a cloth-lined basket, as shown in one of the photos above.
Lastly, if you really want a fix of western fast food, there's a KFC, along with a pizza place and a Dairy Queen, in the Big C supercenter out on Charensuk Road. It's the edge of town but close enough to get to on bicycle.