Why Sarah was angry to Abraham about Hagar

In that time, it was the duty of a wife (and especially of the wife of a man of substance such as Abraham was) to provide her husband with a son and heir. If Sarah could not give Abraham a son, it was her duty to find a woman (her handmaiden, Hagar) who could perform that duty. The child born to the surrogate mother would be legally the child of her mistress/owner (in this case, Sarah).

Nevertheless, Sarah (being a wife and a human being who felt inadequate since she had failed to give her husband an heir) naturally felt resentful and jealous that Hagar had been blessed with a son and she herself had not. She felt jealous of Abraham's kindness to this "other woman" whom Sarah had been forced (at least in her own mind) by conventions of her society to give to Abraham. To compound Sarah's feelings, Hagar began to treat Sarah haughtily since Hagar had bourn a son to Abraham and Sarah had failed to do so.

Following the birth of Isaac, Sarah's own natural son, Ishmael's unkind treatment of this younger half-brother who would now have the inheritance that he and his mother Hagar had confidently expected would be Ishmael's own, made Sarah even more resentful and angry. (Ishmael at that time was no longer a little boy but a teenager.) Sarah blew up and insisted that Hagar and Ishmael be put out of the camp permanently.

Abraham was reluctant to do so, but he was told by God to do what Sarah wanted, and he was assured that God would bless and multiply Ishmael and make him a great patriarch. So Abraham did as God commanded and gave Sarah permission to do as she thought good. Sarah put Hagar and her son out of the camp. You can read the rest of the story. God's promise concerning Ishmael came true, and also His warnings concerning what manner of man Ishmael would become.

Sarah may or may not have been right to be angry with Hagar, but her anger was a very natural and human reaction to the events in her life. She felt trapped, that she had done what was required of her, and she didn't like it. And then Hagar turned up her nose in scorn. It was too much for Sarah to endure.

Many Biblical scholars believe that Sarah was upset that Hagar perceived herself more important for Abraham than Sarah. She wanted to become the "main" wife of Abraham.

(Gen. 16:4) "... When she saw that she had conceived, her mistress was despised in her eyes."

Some commentaries from http://biblehub.com/genesis/16-4.htm:

Ellicott's Commentary for English Readers

"(4) Her mistress was despised.--Hagar, we are told in Genesis 16:3, was to be, not Abram's concubine, but his wife. She was to be Sarai's representative, and though now she would hold the highest place in the household next to Sarai, because of this relation to Abram, yet she would continue to be Sarai's maid. But no sooner had she conceived, than, proud of her superiority over her mistress, she wished to overthrow this arrangement, and, at all events, acted as if she was Abram's wife absolutely, and thrust Sarai aside."

Pulpit Commentary

"Verse 4. - And he went in unto Hagar. בּוא אֶלאּ, a linguistic peculiarity of the Jehovist, occurring Genesis 29:21, 30; Genesis 30:3, 4; Genesis 38:2, 9, 16 (Vaihinger, Davidson); but by some partitionists Genesis 29. and 30. are assigned to the Elohist (Tuch, Bleek, De Wette). And she conceived: and when she saw that she had conceived, her mistress was despised in her eyes. As Hannah by Peninnah (1 Samuel 1:6); barrenness among the Hebrews having been regarded as a dishonor and reproach (Genesis 19:31; Genesis 30:1, 23; Leviticus 20:20), and fecundity as a special mark of the Divine favor (Genesis 21:6; Genesis 24:60; Exodus 23:26; Deuteronomy 7:14). Whether Hagar imagined Sarai to be through her barrenness "tanquam a Divino promisso repudiatam" (Lyra), or anticipated Sarai's displacement from her position as Abram s wife (Inglis), she, immediately on perceiving her condition, became insolent (cf. Proverbs 30:23)."

Gill's Exposition of the Entire Bible

"And he went in unto Hagar, and she conceived,.... The formality of the marriage being over, he enjoyed her as his wife, and she immediately conceived by him:

and when she saw that she had conceived; when she perceived that she was with child:

her mistress was despised in her eyes; she thought herself above her, and treated her as her inferior, with contempt, and reproached her for her barrenness, as Peninnah did Hannah, 1 Samuel 1:6; and it was the more ungrateful, as it was at the motion of her mistress that she was given to Abram for wife."

Matthew Henry Commentary

"16:4-6 Abram's unhappy marriage to Hagar very soon made a great deal of mischief. We may thank ourselves for the guilt and grief that follow us, when we go out of the way of our duty. See it in this case, Passionate people often quarrel with others, for things of which they themselves must bear the blame. Sarai had given her maid to Abram, yet she cries out, My wrong be upon thee. That is never said wisely, which pride and anger put into our mouths. Those are not always in the right, who are most loud and forward in appealing to God: such rash and bold imprecations commonly speak guilt and a bad cause. Hagar forgot that she herself had first given the provocation, by despising her mistress. Those that suffer for their faults, ought to bear it patiently, 1Pe 2:20."