The Bible teaches that physical things related to a holy person can be channels of divine grace and miracles.
Exodus 30:28-29 (KJV) And the altar of burnt offering with all his vessels, and the laver and his foot.  And thou shalt sanctify them, that they may be most holy: whatsoever toucheth them shall be holy. (cf. 29:37; Lev 6:18, 27)
2 Kings 2:13-14 He took up also the mantle of Elijah that fell from him, and went back, and stood by the bank of Jordan;  And he took the mantle of Elijah that fell from him, and smote the waters, and said, Where is the LORD God of Elijah? and when he also had smitten the waters, they parted hither and thither: and Elisha went over.
Elijah’s mantle is an example of a “second-class” relic: items that have power because they were connected with a holy person.
2 Kings 13:21 And it came to pass, as they were burying a man, that, behold, they spied a band of men; and they cast the man into the sepulchre of Elisha: and when the man was let down, and touched the bones of Elisha, he revived, and stood up on his feet.
The bones or relics of Elisha had so much supernatural power or “grace” in them that they could even cause a man to be raised from the dead. His bones were a “first-class” relic: from the person himself or herself.
Mark 5:27-29 When she had heard of Jesus, came in the press behind, and touched his garment.  For she said, If I may touch but his clothes, I shall be whole.  And straightway the fountain of her blood was dried up; and she felt in her body that she was healed of that plague. (cf. Lk 8:43-48)
Jesus did say that the woman’s faith made her well (in the parallel passage Luke 8:48), yet the instrumentality of a physical object in contact with Jesus was also a factor: as indicated precisely by its effect of causing “power” to go “out of him” (Mk 5:30; Lk 8:46). God used the physical object for spiritual (and supernatural physical) purposes: a healing. We see it again, when Jesus heals the blind man:
John 9:6-7 When he had thus spoken, he spat on the ground, and made clay of the spittle, and he anointed the eyes of the blind man with the clay,  And said unto him, Go, wash in the pool of Siloam, (which is by interpretation, Sent.) He went his way therefore, and washed, and came seeing.
Jesus could have simply declared him healed, with no material object used. But instead, He used a bodily fluid (His own), and also clay, or dirt, and then the water of the pool, and rubbed the man’s eyes, to effect the miracle (two liquids, solid matter, and physical anointing action of fingers). The Bible thus teaches that physical things related to a holy person in some fashion can be channels to bring about miracles. This is exactly how Catholics view relics. There are several other examples of the same thing, with touch or matter of some sort being utilized to heal:
Matthew 8:15 And he touched her hand, and the fever left her . . .
Matthew 9:29-30 Then touched he their eyes, saying, According to your faith be it unto you.  And their eyes were opened; . . .
Mark 1:31 And he came and took her by the hand, and lifted her up; and immediately the fever left her, . . .
Mark 7:33, 35 And he took him aside from the multitude, and put his fingers into his ears, and he spit, and touched his tongue; . . .  And straightway his ears were opened, and the string of his tongue was loosed, and he spake plain.
Mark 8:23, 25 And he took the blind man by the hand, and led him out of the town; and when he had spit on his eyes, and put his hands upon him, . . .  After that he put his hands again upon his eyes, and made him look up: and he was restored, and saw every man clearly.
Mark 9:26-27 And the spirit cried, and rent him sore, and came out of him: and he was as one dead; insomuch that many said, He is dead.  But Jesus took him by the hand, and lifted him up; and he arose.
Luke 13:13 And he laid his hands on her: and immediately she was made straight, and glorified God.
See also the examples of lepers healed by Jesus’ touch (Matt. 8:2; Mark 1:40-41; Luke 5:13), and touch used to raise the dead (Matt. 9:24-25; Mark 5:40-42; 8:53-55), and further similar examples in chapter 9. One of these miracles is particularly interesting:
Luke 7:14-15 And he came and touched the bier: and they that bare him stood still. And he said, Young man, I say unto thee, Arise.  And he that was dead sat up, and began to speak. . . .
Note that Jesus merely touched the bier that the coffin was being carried on, not even the person himself. Luke thought that this was important enough to mention. The implication is that grace was indirectly channeled by touch through the bier (an inanimate object) to the dead man, for the purpose of raising him.
Acts 5:15-16: Insomuch that they brought forth the sick into the streets, and laid them on beds and couches, that at the least the shadow of Peter passing by might overshadow some of them.  . . . and they were healed every one.
St. Peter’s shadow is another example of a “second-class” relic. Jesus’ garments and saliva are also in this category.
Acts 19:12 . . . from his [Paul’s] body were brought unto the sick handkerchiefs or aprons, and the diseases departed from them, and the evil spirits went out of them. (cf. Matt. 9:20-22)
St. Paul’s handkerchiefs and aprons are third-class relics: things that have merely touched a holy person or a first-class relic.