How Diseases Work
Each disease has a progression, with positions 0 to 3 or 4; you can imagine it as a horizontal track, like below:
0 | 1 | 2 | 3 | (4?)
When a character is first infected, they start at position 1. If they ever reach position 0, the disease is cured. If they ever reach the rightmost position (4 if present, otherwise 3), the disease has reached its final state and its effects can only be removed by some special means that depends on the disease: usually the Cure Disease ritual works, but sometimes bathing in a specific fountain or some other plot-specific action is required. Endurance and heal checks are no longer made once the disease reaches its final state.
Every disease-specific time increment (extended rest is most common), the infected player must make an endurance check to resist the disease; another character may also make a heal check to try to treat the character (per p185 of the PHB, the infected character uses the highest of their endurance check and any heal checks made on their behalf).
Each disease has two DCs. If the check used is greater than or equal to the high DC, the infected character moves one position to the left on the disease track. If the check used is less than the low DC, the infected character moves one position to the right on the disease track. If the check used is greater than or equal to the low DC but less than the high DC, the infected character does not move on the disease track.
See DMG p49-50 or RC p184-185 for the exact wording and examples.
How to Easily Beat Diseases
Because diseases start only one step from being cured, good results on the first progression check make a huge difference. Beating the high DC on the first try removes the threat entirely, while failing to beat the low DC on the first try means you now have to beat the high DC at least twice to cure the disease.
An ally with heal will generally be able to get better check results than most characters. Wisdom is a relatively common primary attribute, and not unusual as a secondary attribute; most classes with wisdom as their primary attribute get heal as a class skill, as do many of the classes with wisdom as a secondary attribute. Conversely, endurance is not a common choice for a trained skill, and few classes have constitution as their primary or secondary attribute.
Anyone who is trained in heal or endurance and has a 14 in the associated attribute should have a 50% chance to beat the low DC of a same-level disease. A trained character with an 18 in the associated attribute and a +2 skill bonus from their background or race would have a 50% chance to beat the high DC of a same-level disease. These odds only go down by 5% for every 2 levels higher the disease is over the party.
At first glance these numbers may seem unfavorable: only a 50% chance for the average character to not get worse? The trick lies in having multiple characters (especially trained ones) attempt to heal the infected character. Because the highest result is used, infected characters are most likely to end up using a good result for a highly-skilled character or a very good result for an average skill character.
Basically, diseases have to be significantly higher level than the party to present a threat if the party contains even one person with a good heal skill.