The Doppler effect relates to the rate at which events appear to be happening when viewed from some distance, rather than the rate at which they are happening, and the difference between the two rates is caused by the time it takes for the information about what is happening to travel from the source to the observer.
If you are moving towards a planet and you were able to study a clock on it through a powerful telescope as you approached, the clock would appear to be running fast. If you turned round and started moving away from the planet, the clock would seem to be running slow. Clearly the real tick-rate of the clock does not change.
To see why it works this way, suppose I lived a long way from you, and wrote you a letter every day, the letters each taking five days to arrive by post. After an initial delay of five days you will receive my first letter, and after that the letters will arrive daily, as each takes the same time to travel to you. If, however, I started travelling towards you, so that each day that I travel knocks a day off the postage time, then at some point you will receive five letters from me all on the same day. Although I wrote them a day apart, each successive letter has spent a day less in transit, so they all arrive at once. To you, the rate at which I have been writing letters seems to have speeded up.
Conversely, if I had gone travelling in the opposite direction, so each day I added one day more to the postage time, my letters would start arriving with you every two days, even though I was writing them daily, because each successive letter spends a day longer in transit. My rate of latter writing would appear to have slowed down.
My rate of writing letters is always one per day, but the rate at which you receive them is speeded up if I am moving towards you or slowed down if I am moving away.