So, why does the Chief Justice break ties during a impeachment trial and not the Vice President, which is the usual case?
See Does the Vice-President retain their tiebreaking vote during an impeachment trial of the President?.
Also, what authorizes the Chief Justice to cast tie-breaking votes?
Precedent. In the impeachment trial of President Andrew Johnson, the Senate rejected attempts to prevent the Chief Justice from voting.
Procedure and Guidelines for Impeachment Trials in the United States Senate, pp. 40-42.
Chief Justice as Presiding Officer
The Chief Justice has voted in the case of a tie in an impeachment trial on two occasions. On March 31, 1868, a motion was made that the Senate retire for consultation. The yeas were 25 and the nays were 25, and the Chief Justice voted in the affirmative. At this point the Senate retired to its conference chamber.
Various amendments to the impeachment rules were discussed in this conference. As a result of the vote by the Chief Justice, Senator Charles Sumner of Massachusetts moved "That the Chief Justice of the United States, presiding in the Senate on the trial of the President of the United States, is not a member of the Senate, and has no authority, under the Constitution, to vote on any question during the trial, and he can pronounce decision only as the organ of the Senate, with its assent." This was defeated by a vote of 22 yeas to 26 nays. [...]
At the end of the conference Senator Sumner raised the issue of the right of the Chief Justice to vote on any question during the trial, but objection was raised to the fact that this was not germane to the matter on which the Senate had retired to confer and a motion that the Senate return to the Chamber without acting on Senator Sumner’s proposal was agreed to.
During the next day’s proceedings, Senator Sumner again raised the issue of the right of the Chief Justice to vote. During the reading of the Journal, he proposed an amendment to the Journal as follows: "It appearing from the reading of the Journal of yesterday that on a question where the Senate were equally divided the Chief Justice, presiding on the trial of the President, gave a casting vote, it is hereby declared that, in the judgment of the Senate, such vote was without authority under the Constitution of the United States." This was rejected by a vote of 21 yeas, 27 nays. Thus the Senate turned down each attempt to prevent the Chief Justice from voting, and in a subsequent action concerning a motion for adjournment, the vote being yeas 22, nays 22, the Chief Justice voted in the affirmative, deciding the issue. This vote was not challenged. [Emboldening added.]
At the end of the trial of President Johnson, however, another occasion arose on a motion to adjourn to a date certain when the vote was tied 27 to 27 and the Chief Justice refrained from voting.