The A-10 looks the way it does because it is a plane designed not just to fly low, but to fly low, slow, and fully-loaded over terrain containing a lot of people who want to shoot that aircraft down, because they know if they don't do it quickly, they're dead. Every aspect of the A-10's design is intended to facilitate the aircraft's use in such an environment and give it the highest chance of coming back.
The A-10 is actually not intended for deep interdiction; that mission is more often given to the F-15E, F-16, or the B-1B, depending on range, payload requirements and availability. The A-10, especially the newer C variant which has an onboard laser designator for smart bombs, can go bunker-busting behind enemy lines if it has to, but its maximum speed and service ceiling are too low for real penetration work.
The A-10 was built for a single purpose: close air support (or CAS). It was designed to take a lot of weapons into the sky over the front lines, supplementing and supporting friendly infantry and armor by taking out enemy armor and artillery. Most of those enemies would be able to shoot back, using anything from AK-47s and shoulder-fired Stingers to MPAT shells and mobile AAA/SAM launchers like the SA-13 or ZSU-23. To that end, it was built for a high payload capacity, good close-quarters maneuverability, long loiter times, extreme durability, and an on-board cannon capable of engaging armored targets (these last three elements being totally absent from Vietnam-era F-105 and F-4 ground attack aircraft, while Marine Hueys and Cobras had insufficient armor-piercing capability to engage the NVA's Soviet-supplied tanks).
These elements also make the aircraft good for other shorter-range ground missions, such as BAI (Battlefield Air Interdiction; destroying ground forces behind the front lines on their way to make trouble) and SEAD (Suppression of Enemy Air Defenses; this task is more often given to F-16s which can carry the HARM anti-radar missile, but if the A-10 can see a SAM launcher, it can kill it). While not designed for the task, the A-10 is more than a match for enemy helicopters, typically carrying a pair of Sidewinders under one wing (a 30MM burst from the GAU-8 will also do the trick).
It was also built on a budget; with the F-14 and F-15 air superiority programs costing over \$30 million a plane, getting the additional money for a dedicated small ground attack plane through Congress was difficult, so the winner of the AX program had to have a unit cost less than \$1 million. That meant that researching radically new wing planforms and fuselage design concepts just wasn't in the budget; both the A-10 and the YA-9 that lost to it were built very simply using proven design elements hearkening back to the first generation of jet fighters and even to WWII-era light bombers. The A-10A variant originally cost just \$450,000; modern A-10Cs, with glass cockpits and more advanced targeting systems for smart weapons, are still only \$11 million in more recent dollar figures. Compared to the F-15E's \$31 million unit cost, and the A-10's replacement the F-35 costing \$98 million each, the A-10C is still an unbelievable bargain.
As a result of all these design requirements, the A-10 has straight wings with a much thicker cross-section, generating more lift at lower speeds and a smaller target area than the F-15's tennis-court-sized planform. Those thick wings are also designed for 6g maneuvering with up to 16,000 pounds of ordnance hanging from them. An additional reason for so much lift and wing volume is that the aircraft is designed to fly with half a wing missing (in addition to an engine and half its tail), and has dual redundant hydraulic loops to all control surfaces plus a set of good old-fashioned cable controls should the main hydraulic or electrical system fail (the core components for both of these being housed in the "titanium tub" that protects the pilot and core avionics).
The design considerations for the F-15 were different. The requirements for the Eagle favored high speed and maneuverability (the MiG-25 and its purported capabilities had really scared the USAF top brass; it turned out to be fast, but really heavy, and so it wasn't the dogfighter NATO feared it would be when they saw its wings in recon pictures), a high ceiling (it's called the Eagle for a reason) and "not a pound for air to ground", focusing on BVR aerial engagements using Sidewinders and Sparrows (now AMRAAMs) with the additional ability to mix it up at visual range with Sidewinders and guns. In short, the USAF wanted the best pure air-to-air fighter money could buy, and for its day they spared no expense; the F-15C has a unit cost around \$30 million, figured primarily in late 1970s dollars (the equivalent of $133 million a plane today, not that much less than the Raptor program).
The F-16, and its runner-up in the LWF competition the YF-17 (which the Navy would develop into the F/A-18), had a slightly different purpose; to be the best dogfighter that $16 million would buy, that could also drop bombs. The idea was to supplement the ridiculously expensive F-15 with something the USAF could afford to lose a few of in establishing air superiority and suppressing ground-based air defenses. The program emphasized a small, nimble, affordable fighter capable of attacking any target the USAF would be assigned, air or ground.