Shofar arousing Hashem's mercy

בשעה שבנ"י תוקעין בשופר הקב"ה עומד מכסא דין ויושב על כסא רחמים.

At the moment that Bnei Yisrael blow the shofar, the Kadosh Baruch Hu rises from the throne of judgment (kisei din) and sits upon the throne of mercy (kisei rachamim).

The Sefas Emes cites the well-known teaching that the sound of the shofar moves Hashem, as it were, from strict judgment to mercy.

וצריך להיות עיקר המכוון כדי שישב המלך על כסא רחמים.

And the main intention must be that the King should sit upon the throne of mercy.

Our primary kavanah in blowing the shofar should be to bring Hashem, the King, to act with mercy.

כי אין טוב שיגרמו בני אדם שיהי' המלך במדת הדין ע"י.

For it is not good that people should cause, through themselves, that the King be in the attribute of din.

A person should never want his own conduct to be the cause of Hashem relating to the world through strict judgment.

לכן מבקשין בתקיעותינו שיהי' המלך במדת הרחמים וכשעושין זה באמת.

Therefore we request, through our tekios, that the King be in the attribute of rachamim — and when we do this in truth.

Our shofar blasts are a sincere plea that Hashem relate to us with the middah of mercy.

הקב"ה מתמלא רחמים עלינו:

The Kadosh Baruch Hu becomes filled with mercy upon us.

When our plea is genuine, Hashem responds by being filled with compassion toward us.

Summary: The shofar moves Hashem from the throne of din to the throne of rachamim; therefore the essential kavanah of our tekios should be a sincere plea — not to cause judgment through our own deeds, but to bring the King to relate to us with mercy, and when this plea is truthful, Hashem is filled with rachamim upon us.