Existence here connotes our ability to respond to different kind of stimuli. For instance a baby right from her birth starts interacting with the external world. This we can verify by the curious look in her eyes. Babies often stare at any new stimulus such as a toy, a picture, or a person, as if they are trying to make sense of what they see. Sometimes, when a mother smiles at her baby, the baby responds with a smile. This interaction takes place at a very crude level as the mind is still learning to identify all the different kinds of stimuli. In later phases of her life she comes to know a lot more about herself, about her family, her society etc. All this time the interaction with the external world and internal world never stops. She learns to form her own opinions about everything she encounters and acts accordingly.
Thus if she tends to be possessive and do not understand the value of sharing things with others, she may be called a selfish person. This attribute of being selfish has been acquired by her as a consequence of her actions. Later on she may learn the advantages of sharing things with others or helping others thus changing that attribute of being selfish to being a kind, loving and caring person. Thus her attributes which are also called her essence are due to her actions. She is neither selfish nor caring— essentially.
This is how existentialists arrive at the notion that existence that is to think, to choose, and to act comes first, or precedes the attributes or essence acquired by the person.