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What he said:

Dear Brothers and Sisters: We know well that every family on occasion suffers moments when one family member offends another. Through our words, actions, or omissions, instead of expressing love for our spouse or children, we can sometimes diminish or demean that love. Hiding these hurts only deepens such wounds, leading to anger and friction between loved ones. If these wounds are particularly deep, they can even lead a spouse to search for understanding elsewhere, to the detriment of the family, especially children. Being one flesh, any wounds that spouses suffer are shared by their children, born of their flesh. When we remember how Jesus warned adults not to scandalize little ones (cf. Mt 18:6), we better understand the vital responsibility to maintain and protect the bond of marriage which is the foundation of the human family. We thank God that although these wounds may lead some to separation, even then many men and women remain true to their conjugal bond, sustained by faith and by love for their children. For those who enter into so-called irregular situations, we must reflect on how best to help and accompany them in their lives. Let us ask the Lord for a strong faith to see with his eyes the reality of family life, and for a deep love to approach all families with his merciful heart.

Source: General Audience of 24 June 2015

What mainstream media heard/reported:
Yahoo: Pope says family separation can be ‘morally necessary’

which quickly mutated to

New York Post: Pope Francis: Divorce can be ‘morally necessary’
New Zealand Herald: Pope admits divorce may be ‘morally necessary’
Premier Christian Radio: Pope Francis: Divorce sometimes “morally necessary”
from the “Catholic” US Catholic: Can divorce be morally necessary to protect kids? Pope Francis thinks so
Kenya Today: Pope Francis: It is sometimes ‘morally necessary’ to divorce your partner
Australian Women’s Weekly: Pope Francis says divorce is sometimes “morally necessary”
RT.com: Pope Francis: Sometimes divorce can be a ‘moral necessity’

Does mainstream media even do any real reporting anymore?

On the contrary, Pope Francis was emphasizing

When we remember how Jesus warned adults not to scandalize little ones (cf. Mt 18:6), we better understand the vital responsibility to maintain and protect the bond of marriage which is the foundation of the human family.

Even this statement, We thank God that although these wounds may lead some to separation, even then many men and women remain true to their conjugal bond, sustained by faith and by love for their children, can in no way be interpreted as “divorce may be morally necessary”.

It is the same conjugal bond which St. Pope John Paul II talks about in Familiaris Consortio:

Like each of the seven sacraments, so also marriage is a real symbol of the event of salvation, but in its own way. “The spouses participate in it as spouses, together, as a couple, so that the first and immediate effect of marriage (res et sacramentum) is not supernatural grace itself, but the Christian conjugal bond, a typically Christian communion of two persons because it represents the mystery of Christ’s incarnation and the mystery of His covenant. The content of participation in Christ’s life is also specific: conjugal love involves a totality, in which all the elements of the person enter- appeal of the body and instinct, power of feeling and affectivity, aspiration of the spirit and of will. It aims at a deeply personal unity, the unity that, beyond union in one flesh, leads to forming one heart and soul; it demands indissolubility and faithfulness in definitive mutual giving; and it is open to fertility (cf Humanae vitae, 9). In a word it is a question of the normal characteristics of all natural conjugal love, but with a new significance which not only purifies and strengthens them, but raises them to the extent of making them the expression of specifically Christian values.”

The Holy Father isn’t saying anything different from what is in the Catechism of the Catholic Church:

Marriage under the regime of sin

1606 Every man experiences evil around him and within himself. This experience makes itself felt in the relationships between man and woman. Their union has always been threatened by discord, a spirit of domination, infidelity, jealousy, and conflicts that can escalate into hatred and separation. This disorder can manifest itself more or less acutely, and can be more or less overcome according to the circumstances of cultures, eras, and individuals, but it does seem to have a universal character.

1649 Yet there are some situations in which living together becomes practically impossible for a variety of reasons. In such cases the Church permits the physical separation of the couple and their living apart. The spouses do not cease to be husband and wife before God and so are not free to contract a new union. In this difficult situation, the best solution would be, if possible, reconciliation. The Christian community is called to help these persons live out their situation in a Christian manner and in fidelity to their marriage bond which remains indissoluble.

ETA:

Further clarification from Canon Lawyer Ed Peters:

Of course, say “separation” these days and the world hears “divorce”; mention “divorce” and the world immediately assumes a “right to remarriage”. Francis, however, mentions neither divorce nor remarriage. He could have, if he so chose, reminded his audience that even civil divorce (which does not destroy a natural marriage bond, let alone a sacramental one!) is a morally licit option under certain circumstances (see, e.g., CCC 2383), but he would never have suggested a divorced person’s simply proceeding to remarriage, for such would be contrary to the plain teaching of Christ. But civil divorce and remarriage are complex topics that do not lend themselves to adequate presentation to a crowd of pilgrims standing under the hot summer sun.