The Buddha explains this truth by simply listing the various types of Dukkha.

i)  Birth
Birth in a general sense means the entire period of gestation from the time of conception to exit from the womb. Birth in itself, when it takes place becomes a painful experience. Being thrust out from the womb, being thrown out into the world without a choice, without any understanding is a traumatic experience. Birth is dukkha also since this is the first point for all other forms of Dukkha that will follow during the course of life. After birth growth takes place, which also has its share of pain.
ii)  ageing
When the maximum point of growth is reached, ageing sets in. The skin wrinkles, the teeth begin to fall out, sense faculties loose their sharpness, hair turns grey, memory fades and vitality declines.
iii)  Disease
Sickness whether physical or mental is suffering.
iv)  Death
At the end comes death. The break up of the body, the extinguishing of the life force is suffering.
v)  Sorrow, lamentation, pain , grief and despair
Sorrow is intense woe because of some deprivation.
Lamentation is crying and weeping.
Pain is bodily pain.
Grief is any kind of mental unhappiness.
Despair is the lowest point of mental anguish, when all hope is given up.
vi)  Union with the unpleasant
Facing the various unpleasant situations and disagreeable people we don't want to face is suffering, when we are thrown into them against our will.
vii)  Separation from the pleasant
There are pleasant and agreeable situations or people we would like to meet with and hope will last, or we want to cling or hold to or relationships we want to endure. Facing separation from these pleasant situations or people is suffering.
viii)  Not to get what we desire
Generally we desire pleasure, wealth, fame and praise, but instead one meet with pain, poverty, dishonour and blame. When we want to remain young, we grow old, when we want to be healthy we fall sick. All this is suffering.

Then the Buddha sums up: "In brief the five aggregates of clinging are dukkha". With this statement the Buddha indicates that all our experience is included in dukkha. The five aggregates are the basic components of our experience, which is of five types. They are material form, feeling, perception, mental formations and consciousness.

The material form covers the physical body with its sense faculties and the other four are the mental side. The reason they are all included in Dukkha is that they are all impermanent, changing from moment to moment. In fact they are only momentary events without any inner core. What we call "my self" is just a combination of aggregates changing from moment to moment. It is the aggregates that are born, that grow old and finally die.

Suffering in Depth
Is Buddha a Pessimist?