The problem that many parents have is that they cannot bear if their child is unhappy or behaves "inappropirately" (e.g. tantrums). The problem is not that a child feels sad or angry or whatever, but that the parent cannot differentiate their own person and feelings from those of the child and that many adults live in a world where any expression of feelings apart from happiness is inappropriate and where any state except happiness is seen as failing at life. These parents feel the sadness of their child as their own sadness and they are ashamed of their child's behavior as if they themselves had behaved in that way.
All children go through phases where they throw themselves on the floor and wail, and all children grow out of that behavior by themselves and without any adult intervention. It happens to everyone and **it passes*. Nevertheless many parents feel as if only their child threw tantrums and as if it was a failure on their part. (This feeling is enforced by the growing number of adults that do not and will never have children of their own and perceive anything that a child does beyond sitting quietly as an intolerable transgression.)
The advice to not talk children out of how they feel tells parents to not identify with their children and accept that they are their own person. People are sad and angry and so on from time to time, and it is normal and healthy to feel that way in certain situations. In fact, as Adam Davis has pointed out in his answer, it is unhealthy to not allow yourself or your child to have these emotions. A child that suppresses their tears or rage or melancholy for the sake of their parent will group up into a dysfuctional adult. If you let your child feel how he or she feels without interfering, they learn how to regulate their emotions. If you try to help them not to feel that way, no matter how sensitive you think you are about it, your child cannot learn this one single most important cultural technique. Psychological research has shown that people who regulate their emotions better are less prone to xenophobia and interpersonal violence.
So if your child cries, allow them to cry until they have run out of tears. Talk to them about it only after they have calmed completely, maybe even after a few hours or on the next day.
Just think about what you yourself would want if you felt sad. Does it help if someone say "it's not so bad"? Does it help, if they want to discuss why you feel sad and what you might do to avoid feeling sad? No! What you want is for your friend to keep their mouth shut (or make soothing noises) and otherwise hold you while you cry until you are done. And then you will feel good and feel happiness returning in a quite wondrous way, without anyone doing anything at all, except allowing you to cry.
Do the same for your child.