This famous saying was coined in Pembrokeshire (Wales) in the mid 19th century, with other similar phrases such as "Eat an apple on going to bed, And you'll keep the doctor from earning his bread" or "Ait a happle avore gwain to bed, An' you'll make the doctor beg his bread".

I've read a post about the benefits of eating apples and I'm going to sum up the 8 good points mentionned in such text:

1- Apples contain Vitamin C, which helps greatly our immune system.
2- They contain flavonoids which, as well as having antioxidant effects, prevent both coronary heart disease and cardiovascular disease.
3- They're low in calories compared with any other sweet product (only 70-100 calories), but they contain sugar alike, so the desire to eat sweets disappear when taking an apple.
4- They help prevent cancers (colon, prostate and breast).
5- As this fruit contains phenols, it reduces bad cholesterol and increases good cholesterol.
6- They prevent tooth decay, produced by a bacteria that dissappears in a very high proportion (80 %) thanks to the liquid apples had inside them.
7- In addition, due to the phytonutrients they contain, apples prevents neurodegenerative disorders (say Alzheimer's and Parkinsonism).
8- Eating 5 apples a week reduces breathing problems such as asthma.

So, we can understand why it's claimed that eating apples is so healthy.