The EXperiment for Cryogenic Large-Aperture Intensity Mapping (EXCLAIM) is a balloon-borne far-infrared telescope that will survey star formation history over cosmological time scales to improve our understanding of why the star formation rate declined at redshift z < 2, despite continued clustering of dark matter. Specifically, EXCLAIM will map the emission of redshifted carbon monoxide and singly-ionized carbon lines in windows over a redshift range 0 < z < 3.5, following an innovative approach known as intensity mapping. Intensity mapping measures the statistics of brightness fluctuations of cumulative line emissions instead of detecting individual galaxies, thus enabling a blind, complete census of the emitting gas. To detect this emission unambiguously, EXCLAIM will cross-correlate with a spectroscopic galaxy catalog. The EXCLAIM mission uses a cryogenic design to cool the telescope optics to approximately 1.7 K. The telescope features a 90-cm primary mirror to probe spatial scales on the sky from the linear regime up to shot noise-dominated scales. The telescope optical elements couple to six µ-Spec spectrometer modules, operating over a 420-540 GHz frequency band with a spectral resolution of 512 and featuring microwave kinetic inductance detectors. A Radio Frequency System-on-Chip (RFSoC) reads out the detectors in the baseline design. The cryogenic telescope and the sensitive detectors allow EXCLAIM to reach high sensitivity in spectral windows of low emission in the upper atmosphere.