John 4:23-24
23 “But an hour is coming, and now is, when the true worshipers will worship the Father in spirit and truth; for such people the Father seeks to be His worshipers. 24 “God is spirit, and those who worship Him must worship in spirit and truth.”
I take this to mean a sort of communion (not in the Catholic sense) with human spirit and God who is spirit.
Luke 17 (KJV)
And when he was demanded of the Pharisees, when the kingdom of God should come, he answered them and said, The kingdom of God cometh not with observation: Neither shall they say, Lo here! or, lo there! for, behold, the kingdom of God is within you.
I take this to mean that when you have this "communion" with God, you do so "without observation", invisibly and in secret.
Matt. 6:6 - But thou, when thou prayest, enter into thy closet, and when thou hast shut thy door, pray to thy Father which is in secret; and thy Father which seeth in secret shall reward thee openly.
Remember Jesus condemned the Pharisees for praying in public so as to be notice by others as being pious.
Matt. 6:5 - And when thou prayest, thou shalt not be as the hypocrites are: for they love to pray standing in the synagogues and in the corners of the streets, that they may be seen of men. Verily I say unto you, They have their reward.
So when someone says they are spiritual. I take it to mean they have an inner belief in a higher power, but do not subscribe to any particular dogma or church.