My dryer smelled like it was burning so I took it apart, cleaned it out and put a new belt on. That didn't seem to fix the problem so I took it apart again and took the motor out. The shaft was so hard to turn so I assumed that I needed a new motor. I went and put a new motor in it and noticed that there was a black wire that was previously attached to the left lower side of the heat housing. It looks like it connects to the heating element. The wire looked like it was burnt so i put some electrical tape on it and plugged it back on. Put everything back together and it worked on one load of clothes but when I took those out they seemed too hot( it didn't seem normal). Put another load in and checked on those and there was no heat at all. Took it apart again and noticed that the black wire was burnt again leaving the metal fastener still attached to the plug and the wire was completely unattached. I don't know what to do, can you help me, please?

Here are your parts
Replacement parts for MAYTAG MDE7500AYW DRYER- ELE | AppliancePartsPros.com

See the attachment for a manual.
16023110 Maytag Sloped Front Dryer Repair Service Manual - ApplianceDigest.com
You will have to join the site to download the manual but it is free and has lots of info for the appliance DIYer.

Unplug the unit and check the cycling thermostat and the high-limit thermostat with a meter both should be 0 ohms.

Then check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it so the thermal cut-off blows.
This may have contributed to the wire burning out.
This may have been your problem and it has now burned out the heating coil or a thermostat.

Now for your original problem.
Just taping up the wire would not have fixed the problem as you have seen.
If you have a dirty or pitted connection it is a resistance. So when current passes through it heat is generated basically that is how a heating coil works.
This causes more pitting etc., which increases the resistance which increases the heat and so on and so on till the connection fails.

You will have to cut back the wire till you get to nice clean copper and the attach a new connector. Be sure to use one rated for high current.

From the sound of it I would also replace the heating relay.

If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.

Sorry to be a bother but what number is the cycling thermostat on the parts list. I found the hi-limit thermostat but I don't see where the cycling thermostat is. Thank you again