WotC cites the overwhelming prevalence of Deathrite Shaman as a "four of" in more than half of the format's top decks as a primary reason for the card's ban. Decks featuring the card had a reported 55% win rate against the field and eleven of the top sixteen decks ran playsets of it in the recent Grand Prix Birmingham. WotC points out, however, that that's not where their reasoning stops.
"While a card seeing lots of competitive play doesn't by itself necessitate action, in this case we also see a reduction in diversity in the environment, an inability for the metagame to adjust, and a dominant strategy that's particularly hostile to rogue decks and innovation," writes WotC's Ian Duke. "For these reasons, Deathrite Shaman is banned in Legacy."
The other newly-banned card, Gitaxian Probe, has had its share of trouble lately. Already disallowed in Modern and restricted to one copy per deck in Vintage, the card's banning in Legacy is largely due to it giving players too much of an advantage at too low of a cost.
"It's been a main component of the success of the two most played Legacy decks on Magic Online, Grixis Delver and Ad Nauseam Tendrils," mentions Duke. "Gitaxian Probe adds significant power to these decks and others by quickly filling the graveyard and counting toward abilities that require casting spells or drawing cards, without requiring mana investment."
WotC says they wanted to remove the card from the Legacy format before things reached the "boiling point."
Looking forward, it appears that the recently-printed Goblin Chainwhirler may be on WotC's radar for a possible banning in Standard at some point in the future, but the company says they want to see how the soon-to-release Magic Core Set 2019 will affect the format before decided whether or not to do so.